import hashlib
import base64

def mystery(x):
    """
    This function takes a value x, subjects it to a series of transformations,
    and returns True only if the final result matches a secret value.
    """
    # The input must be an integer to proceed.
    if not isinstance(x, int):
        return False

    # Transformation 1: Convert the integer to its hexadecimal string representation.
    # For example, 212652 becomes '0x33ea4'.
    s1 = hex(x)

    # Transformation 2: Take the hex string, remove the '0x' prefix, and reverse it.
    # For example, '33ea4' becomes '4ae33'.
    s2 = s1[2:][::-1]

    # Transformation 3: Encode the reversed string into bytes and then apply Base64 encoding.
    # For example, '4ae33' becomes b'NGFlMzM='.
    s3 = base64.b64encode(s2.encode('ascii'))

    # Transformation 4: Compute the SHA-1 hash of the Base64 encoded bytes.
    # The result is a hex digest string.
    h = hashlib.sha1(s3).hexdigest()

    # The final check: does the computed hash match the secret target hash?
    # It is computationally infeasible to reverse this step.
    return h == '67ae1a64661ac8b4494666f58c4822408dd0a3e4'

import cmath

def mystery(x):
    """
    This function takes a value 'x', interprets it as coordinates
    in a 2D plane, and simulates a process. It returns True only if
    the process terminates in a very specific state at a specific time.
    """
    # --- Input Validation and Parsing ---
    # The input must be a string representing two comma-separated numbers.
    if not isinstance(x, str):
        return False
    
    parts = x.split(',')
    if len(parts) != 2:
        return False
        
    try:
        # These parts define a point 'c' in the complex plane.
        c = complex(float(parts[0]), float(parts[1]))
    except (ValueError, TypeError):
        return False

    # --- The Core Simulation ---
    # We start a process at the origin (z = 0) and repeatedly apply a
    # transformation involving the input point 'c'.
    z = complex(0, 0)
    max_steps = 100
    
    for i in range(1, max_steps + 1):
        z = z*z + c
        
        # The process can become unstable and "escape". We check for this.
        if abs(z) > 2:
            
            # First constraint: The escape must happen at a precise moment.
            # Not sooner, not later.
            if i != 88:
                return False
            
            # Second constraint: The state of the system ('z') at the moment
            # of escape must have a very specific property, checked to a
            # high degree of precision. This makes finding the solution by
            # chance extremely unlikely.
            target_value = 1.304881023
            tolerance = 1e-9
            
            if abs(z.real - target_value) < tolerance:
                return True
            else:
                return False
    
    # If the process remains stable and never escapes, it's not the solution.
    return False

import hashlib

def mystery(x):
    """
    This function takes a string x, uses it to generate a deterministic
    random walk, and returns True only if the walk's path has a set of
    very specific, rare properties.
    """
    # --- Input Validation ---
    # The input must be a string to serve as a seed.
    if not isinstance(x, str):
        return False

    # --- Seeding the Process ---
    # The input string is hashed to produce a starting seed for a
    # pseudo-random number generator (PRNG). This ensures that the same
    # input will always produce the same path, but even a tiny change
    # to the input will produce a completely different path.
    h = hashlib.sha256(x.encode('utf-8')).digest()
    seed = int.from_bytes(h[:8], 'big')

    # --- The Walker Simulation ---
    # The PRNG is a simple Linear Congruential Generator (LCG).
    # Its state is updated in a deterministic sequence based on the seed.
    # Parameters are from C++11's minstd_rand.
    a = 48271
    m = 2**31 - 1
    state = seed % m
    
    # The walker starts at (0,0) on a 16x16 grid and takes 256 steps.
    pos = [0, 0]
    path = [(0, 0)]
    visited = {(0, 0)}
    
    for _ in range(256):
        # Generate the next pseudo-random number.
        state = (a * state) % m
        
        # Use the two least significant bits to determine the direction.
        direction = state & 3
        if direction == 0:   # North
            pos[1] += 1
        elif direction == 1: # East
            pos[0] += 1
        elif direction == 2: # South
            pos[1] -= 1
        else:                # West
            pos[0] -= 1
        
        # Constraint 1: The walker must stay within the 16x16 grid.
        # If it ever steps out of bounds, the path is invalid.
        if not (0 <= pos[0] < 16 and 0 <= pos[1] < 16):
            return False
        
        path.append(tuple(pos))
        visited.add(tuple(pos))
    
    # --- Final Property Checks ---
    # After the walk is complete, the generated path must satisfy three
    # highly specific and independent constraints.

    # Constraint 2: The walker's final position must be exactly (1, 7).
    if tuple(pos) != (1, 7):
        return False

    # Constraint 3: The number of unique grid squares visited during the
    # entire walk must be exactly 111.
    if len(visited) != 111:
        return False

    # Constraint 4: A complex checksum, calculated from the coordinates of
    # every single step in the path, must match a secret target value.
    # This ensures the entire path shape is correct, not just its endpoints.
    checksum = 0
    for i, (px, py) in enumerate(path):
        # The formula is arbitrary but sensitive to the path's structure.
        checksum += (px * (i + 1)) - (py * (i + 1)**2)
    
    if checksum != -11823774:
        return False
        
    # Only a seed that produces a path satisfying all constraints is a solution.
    return True

def mystery(x):
    """
    This function takes a binary string 'x', uses it as the initial
    state for a 1D cellular automaton (Rule 110), and simulates its
    evolution. It returns True only if the state after 128 generations
    exhibits a specific, rare combination of properties.
    """
    # --- Input Validation ---
    if not isinstance(x, str) or len(x) != 64:
        return False
    if not all(c in '01' for c in x):
        return False

    # --- Cellular Automaton Simulation ---
    # The rule is Rule 110, a known chaotic and Turing-complete rule.
    # The keys represent the 3-cell neighborhood (left, center, right).
    rules = {
        (1, 1, 1): 0, (1, 1, 0): 1, (1, 0, 1): 1, (1, 0, 0): 0,
        (0, 1, 1): 1, (0, 1, 0): 1, (0, 0, 1): 1, (0, 0, 0): 0,
    }

    # Convert the input string to a list of integers.
    current_gen = [int(c) for c in x]
    width = len(current_gen)
    num_generations = 128

    for _ in range(num_generations):
        next_gen = [0] * width
        for i in range(width):
            # Get neighborhood with wrap-around boundaries.
            left = current_gen[i - 1]
            center = current_gen[i]
            right = current_gen[(i + 1) % width]
            
            neighborhood = (left, center, right)
            next_gen[i] = rules[neighborhood]
        current_gen = next_gen

    # The final state of the automaton after all generations.
    final_gen_str = "".join(map(str, current_gen))

    # --- Final Property Checks ---
    # The final generation must satisfy three specific and independent constraints.
    # Finding an initial state 'x' that leads to this exact outcome is
    # computationally infeasible without knowing the original 'x'.

    # Constraint 1: The population count (number of '1's) must be exactly 23.
    if final_gen_str.count('1') != 23:
        return False

    # Constraint 2: The final string must contain the specific substring '100001'.
    # This is a rare pattern that constrains the local structure.
    if '100001' not in final_gen_str:
        return False

    # Constraint 3: A weighted checksum of the final state must match a target value.
    # This ensures the global structure of the entire string is correct.
    checksum = sum(i * int(c) for i, c in enumerate(final_gen_str))
    if checksum != 815:
        return False

    return True

import hashlib

def mystery(x):
    """
    This function takes a string 'x' which it treats as a key. It uses this
    key to decrypt a block of data. It returns True only if the decrypted
    data has a very specific, complex internal structure.
    """
    # --- Input Validation ---
    # The key must be a non-empty string.
    if not isinstance(x, str) or not x:
        return False

    # --- The Encrypted Data Block ---
    # This is the "treasure map" that must be decrypted. In its current
    # form, it is indistinguishable from random noise.
    encrypted_map = b'\x02\x1a\x10\xc1s\xe0\x1f\xe3\x91f\xe7G\x86\xf5R\xac\xc3\xb5\x85\xe8\xe2\xd4\xdc\xe8\xc4\xa1\x9d\xe6\xcfF\xd9\x1e\xf5!\xc9\x06\x1a\x96\xb3\x8f\x03\xe8\xc9\x90\xe1\xd2\x15\xe3\x85a\xec\xbc\xd1\x8c\xfc;\xe7\x89\x1f\xee*\x9b\xe9*\xc8\x10\xe6\x17\xd2/\x96\xba\x81\x0e\xc5\x9f\xea\xd2\x05\xea\x8e`\xee\xb3'

    # --- Key Derivation and Decryption ---
    # The input string 'x' is hashed to create a 256-bit (32-byte) key.
    # This is a one-way process; you cannot get 'x' back from the key.
    key = hashlib.sha256(x.encode('utf-8')).digest()
    
    # A simple XOR cipher is used to decrypt the map. A different key
    # will produce completely different, seemingly random, output.
    try:
        decrypted_bytes = bytearray()
        for i in range(len(encrypted_map)):
            decrypted_bytes.append(encrypted_map[i] ^ key[i % len(key)])
        
        # The plaintext is expected to be ASCII text.
        decrypted_map_str = decrypted_bytes.decode('ascii')
    except UnicodeDecodeError:
        # If the key is wrong, the output is likely not valid ASCII.
        return False

    # --- Structural Validation of the Decrypted Data ---
    # The decrypted data is only considered valid if it represents a
    # solved 9x9 Sudoku puzzle. This requires satisfying multiple
    # stringent constraints simultaneously.
    
    # Constraint 1: Must be exactly 81 characters, all of which are digits.
    if len(decrypted_map_str) != 81 or not decrypted_map_str.isdigit():
        return False

    try:
        grid = [[int(decrypted_map_str[r*9 + c]) for c in range(9)] for r in range(9)]
        # A solved Sudoku contains only digits 1-9.
        if any(0 in row for row in grid):
            return False
    except (ValueError, IndexError):
        return False

    # Constraint 2: Each row and each column must contain the digits 1-9 exactly once.
    for i in range(9):
        if len({grid[i][j] for j in range(9)}) != 9: # Check row i
            return False
        if len({grid[j][i] for j in range(9)}) != 9: # Check column i
            return False

    # Constraint 3: Each of the nine 3x3 subgrids must contain the digits 1-9 exactly once.
    for box_r in range(0, 9, 3):
        for box_c in range(0, 9, 3):
            box_vals = {grid[r][c] for r in range(box_r, box_r + 3) for c in range(box_c, box_c + 3)}
            if len(box_vals) != 9:
                return False
    
    # If the decrypted data successfully passes all structural checks, the key 'x' must be correct.
    return True