from math import prod, sqrt
from pathlib import Path
from typing import TypeAlias

Point: TypeAlias = tuple[int, int, int]
Circuit: TypeAlias = set[Point]

FILE = "input.txt"

def main():
    lines = Path(FILE).read_text().splitlines()
    points: list[Point] = [parse_point(line) for line in lines]
    distances = sorted([
        (distance(p1, p2), Circuit([p1, p2]))
        for i, p1 in enumerate(points)
        for p2 in points[i+1:]
    ], key=lambda x: x[0])
    return prod(
        p[0]
        for p in last_circuit(
            [c for _, c in distances], len(points)
        )
    )


def last_circuit(circuits: list[Circuit], total: int) -> Circuit:
    all: list[Circuit] = []
    for circuit in circuits:
        connected = [
            a
            for a in all
            if circuit & a
        ]
        all = [a for a in all if a not in connected]
        all.append(Circuit.union(circuit, *connected))
        if len(all) == 1 and len(all[0]) == total:
            return circuit
    assert False, "unreachable"


def distance(p1: tuple[int, int, int], p2: tuple[int, int, int]):
    x1, y1, z1 = p1
    x2, y2, z2 = p2
    return sqrt((x2-x1)**2+(y2-y1)**2+(z2-z1)**2)


def parse_point(line: str) -> Point:
    x, y, z = line.split(",")
    return (int(x), int(y), int(z))


if __name__ == "__main__":
    print(main())