import base64
import zlib

def mystery(x):
    """
    A puzzle that requires decoding a hidden message and satisfying a mathematical condition.
    The input x must be a string or a number.
    Returns True for a very specific set of inputs, False otherwise.
    """
    # Stage 1: The input must be a string representation of a number.
    if not isinstance(x, str) or not x.isnumeric():
        return False

    # Stage 2: The number must satisfy a seemingly arbitrary mathematical property.
    # This acts as a filter and a potential misdirection.
    n = int(x)
    if n % 999 != 730:
        return False

    # Stage 3: The core of the puzzle lies in this opaque data block.
    # What could it be?
    try:
        data = b'eJzLKCkpKLbS10/LT0m1U9JRcslwK8pLzE1VslLwzEnVLUvMycxLVdJRSs5LzE21UgguSExOVdJRSi5xLC4pSk3MylZSSsovTkxOVQguSSxJ1VFKLi0pycxN1QEA1/kjLg=='
        
        # A series of transformations are applied. The final result is a list.
        # The ultimate test is whether the input 'x' is a member of this hidden list.
        secret_list = zlib.decompress(base64.b64decode(data)).decode().split(',')
        
        return x in secret_list
    except Exception:
        # If any part of the decoding fails, the puzzle remains unsolved.
        return False

import hashlib

def mystery(x):
    """
    A puzzle of fate and numbers. Can you find the origin?
    Only integers of a certain... calibre... need apply.
    """
    # Stage 1: Basic qualifications. Not just any number can be a progenitor.
    if not isinstance(x, int) or isinstance(x, bool):
        return False

    # A curious, almost numerological, constraint. A red herring or a deep clue?
    if sum(int(d) for d in str(x)) <= ((ord('z') - ord('a') + 1) * 2 - 2):
        return False

    # Stage 2: The constants of this universe.
    # They are fixed, but from where do they arise?
    A = (ord('F') - ord('A'))
    B = (ord('B') - ord('A'))
    N_STEPS = len(hashlib.sha1(b"a_stitch_in_time").hexdigest()) + 10

    # Stage 3: The Destination. All paths are measured against this.
    # A value derived from ancient, forgotten words.
    s1 = b"primordial_soup"
    s2 = b"cosmic_background"
    h1 = hashlib.sha512(s1).digest()
    h2 = hashlib.sha512(s2).digest()
    target_val = int.from_bytes(bytes(a ^ b for a, b in zip(h1, h2)), 'big')
    TARGET = target_val % (10**20) + 1337

    # Stage 4: The Journey.
    # A number's evolution through time. Can you reverse the flow?
    n = x
    for _ in range(N_STEPS):
        if n % 2 == 0:
            n = n // 2
        else:
            n = A * n + B
    
    return n == TARGET

import struct
import re

def mystery(x):
    """
    To solve this puzzle, you must command a very simple machine.
    Your input is not a value to be tested, but a program to be executed.
    Find the sequence of operations that leads to the destined state.
    """
    # --- Stage 1: The Constants of this Universe ---
    # These values are fixed. Their origin is obscure, but their purpose is clear.
    # The solver must first understand what these bytes represent.
    DATA = b'\x93\x18\x1b\x9e\xd1\xe4\xe8\xc8\x8f\x8c\x1d\xef\x0f\x8f\x9a\x9b\xe9\x9f\x18\xb1'
    TARGET_A = -7849900015792942736

    # The format string is a crucial clue.
    try:
        k, target_b, target_c, m_extra = struct.unpack('>qiiI', DATA)
    except struct.error:
        # This part of the universe is stable. This should not fail.
        return False

    # --- Stage 2: The Virtual Machine ---
    # The machine has three general-purpose registers: A, B, C.
    # It starts in a known state derived from the constants.
    if not isinstance(x, str):
        return False

    regs = {'A': k, 'B': 0, 'C': 0}
    
    # --- Stage 3: The Program and its Execution ---
    # The program is a sequence of instructions separated by semicolons.
    # The instruction set is small: MOV, ADD, SUB, XOR, HALT.
    # Format: "OPCODE DEST,SOURCE" e.g., "MOV A, 123" or "ADD B, C"
    try:
        instructions = [s.strip() for s in x.split(';') if s.strip()]
        if not instructions or len(instructions) > 10: # Program must be concise.
             return False
    except Exception:
        return False
        
    ip = 0
    max_steps = 20 # The machine has a short lifespan to prevent paradoxes.
    
    for _ in range(max_steps):
        if not (0 <= ip < len(instructions)):
            break # Program finished or jumped out of bounds.
        
        line = instructions[ip]
        ip += 1

        if line.upper() == 'HALT':
            break

        parts = re.split(r'[,\s]+', line, maxsplit=2)
        if len(parts) != 3:
            return False # Invalid instruction format.

        op, r1_str, v2_str = parts[0].upper(), parts[1].upper(), parts[2]

        if r1_str not in regs:
            return False # Invalid destination register.

        # The second operand can be a register or a literal integer.
        val2 = 0
        if v2_str.upper() in regs:
            val2 = regs[v2_str.upper()]
        else:
            try:
                val2 = int(v2_str)
            except ValueError:
                return False # Invalid literal.

        # Execute the instruction.
        if op == 'MOV':
            regs[r1_str] = val2
        elif op == 'ADD':
            regs[r1_str] += val2
        elif op == 'SUB':
            regs[r1_str] -= val2
        elif op == 'XOR':
            regs[r1_str] ^= val2
        else:
            return False # Unknown instruction.

    # --- Stage 4: The Final Judgement ---
    # After the program halts (or runs out of steps), the machine's state
    # is compared against the target state.
    final_state = (regs['A'], regs['B'], regs['C'])
    target_state = (TARGET_A, target_b, target_c)

    return final_state == target_state

import hashlib

def mystery(x):
    """
    Your input is a series of commands to navigate a hidden landscape.
    Find the correct path to reach the destination.
    The landscape is vast and unforgiving; brute force is not the way.
    """

    # --- Stage 1: Input Validation ---
    if not isinstance(x, str) or not all(c in 'NSEW' for c in x):
        return False
    
    # A journey too long or too short will fail.
    if not (10 < len(x) < 50):
        return False

    # --- Stage 2: Defining the Universe ---
    # The constants that shape the landscape are derived, not explicit.
    p1 = b"the_cartographers_secret"
    p2 = b"the_navigators_dilemma"
    
    h1 = hashlib.sha256(p1).digest()
    h2 = hashlib.sha256(p2).digest()
    
    # The landscape's fundamental properties (seed, dimensions)
    seed = bytes(a ^ b for a, b in zip(h1, h2))
    GRID_SIZE = 128 # The world is a 128x128 torus

    # The start and end points of the journey are fixed points in this universe.
    start_row = int.from_bytes(h1[:2], 'big') % GRID_SIZE
    start_col = int.from_bytes(h1[2:4], 'big') % GRID_SIZE
    
    target_row = int.from_bytes(h2[:2], 'big') % GRID_SIZE
    target_col = int.from_bytes(h2[2:4], 'big') % GRID_SIZE
    
    # --- Stage 3: The Laws of Physics (is this a wall?) ---
    # A cell's nature is determined by a cryptographic hash.
    # This makes the landscape unpredictable without local probing.
    def is_wall(r, c):
        hasher = hashlib.sha256()
        hasher.update(seed)
        hasher.update(f"({r},{c})".encode())
        # A simple but arbitrary rule: wall if the last byte's LSB is 1
        return hasher.digest()[-1] & 1 == 1

    # Ensure the start and target points are not themselves walls.
    # This is a critical property for a solvable maze.
    if is_wall(start_row, start_col) or is_wall(target_row, target_col):
        # This branch indicates a paradox in the universe's creation.
        # It should not be reachable in a well-formed puzzle.
        return False

    # --- Stage 4: The Journey ---
    # Simulate the path taken based on the input commands.
    row, col = start_row, start_col
    
    for move in x:
        dr, dc = {'N': (-1, 0), 'S': (1, 0), 'E': (0, 1), 'W': (0, -1)}[move]
        
        # The world wraps around (toroidal grid).
        next_row = (row + dr) % GRID_SIZE
        next_col = (col + dc) % GRID_SIZE
        
        if is_wall(next_row, next_col):
            # The path is blocked. The journey ends in failure.
            return False
        
        row, col = next_row, next_col

    # --- Stage 5: The Destination ---
    # Did the journey end at the correct location?
    return (row, col) == (target_row, target_col)

import hashlib
import math

def mystery(x):
    """
    To solve this puzzle, you must uncover a secret number and reveal its
    primordial components. The path is through cryptography and number theory.
    """

    # --- Stage 1: The Primality Test Sub-routine ---
    # A necessary tool for this universe. Is it a clue, or just a constraint?
    def is_prime(n):
        """
        A standard Miller-Rabin primality test. It is probabilistic for very
        large numbers but deterministic for the numbers relevant to the solution.
        """
        if not isinstance(n, int) or n < 2: return False
        if n == 2 or n == 3: return True
        if n % 2 == 0 or n % 3 == 0: return False
        
        d, s = n - 1, 0
        while d % 2 == 0:
            d //= 2
            s += 1
        
        # A fixed set of bases for deterministic behavior on inputs of this scale.
        bases = [2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37]
        if n < 341550071728321: # Deterministic for n up to this value
            bases = [p for p in bases if p < n]

        for a in bases:
            x = pow(a, d, n)
            if x == 1 or x == n - 1:
                continue
            for _ in range(s - 1):
                x = pow(x, 2, n)
                if x == n - 1:
                    break
            else:
                return False
        return True

    # --- Stage 2: Input Parsing and Validation ---
    # The input must be a comma-separated string of numbers.
    if not isinstance(x, str):
        return False
    
    try:
        parts = [p.strip() for p in x.split(',')]
        if not parts or any(not p.isnumeric() for p in parts):
            return False
        factors = [int(p) for p in parts]
    except (ValueError, AttributeError):
        return False

    # The secret number is composite. A single prime is not the answer.
    if len(factors) < 2:
        return False

    # Every provided number must be prime. This is a crucial constraint.
    for f in factors:
        if not is_prime(f):
            return False

    # --- Stage 3: The Hidden Number ---
    # The target number is derived from the SHA-512 hashes of two phrases.
    # The solver must replicate this process to find the number to be factored.
    s1 = b"the_lonely_mountain"
    s2 = b"the_hidden_door"
    h1 = hashlib.sha512(s1).digest()
    h2 = hashlib.sha512(s2).digest()
    
    target_e = int.from_bytes(bytes(a ^ b for a, b in zip(h1, h2)), 'big')

    # --- Stage 4: The Final Check ---
    # The product of the provided prime factors must equal the hidden number.
    product = 1
    for f in factors:
        product *= f
        
    return product == target_e