#include <iostream>
#include <string>
#include <functional>
#include <vector>

// APEX Math Engine
// Purpose: Validates computational logic output
// License: Apache 2.0

using namespace std;

// A simple simulated "Hard Math" check
// In production, this would verify specific NP-Hard problem solutions
bool verify_computation(string puzzle_hash, string user_solution) {
    // For Vibe Coding: We check if the hash of the solution matches criteria
    hash<string> hasher;
    size_t hash_val = hasher(user_solution + puzzle_hash);
    
    // Simple difficulty check: Hash must be even (50% chance for demo)
    // Real APEX uses specific difficulty targets
    return (hash_val % 2 == 0); 
}

int main(int argc, char* argv[]) {
    if (argc < 3) {
        cout << "Usage: ./apex_engine [puzzle_id] [solution]" << endl;
        return 1;
    }

    string puzzle = argv[1];
    string solution = argv[2];

    if (verify_computation(puzzle, solution)) {
        cout << "VALID" << endl;
        return 0; // Success
    } else {
        cout << "INVALID" << endl;
        return 1; // Failure
    }
}