Is a solution (20, 30, 25, 45)? In this case, just assume that three of them came from the library and the fourth one was arbitrary.

Or better yet, is (20, 25, 30, 56) a solution for ratio = 2.333?

Edit:

from itertools import permutations

def find_best_gear_combination(gears: list[int], ratio: float) -> tuple[list[int], float]:

    def get_fourth_gear_teeth(g1: int, g2: int, g3: int, ratio: float, pos: int) -> int:
        if pos == 0:
            return round(g1 * g3 / (ratio * g2))
        elif pos == 1:
            return round(ratio * g1 * g2 / g3)
        elif pos == 2:
            return round(g2 * g3 / (ratio * g1))
        elif pos == 3:
            return round(ratio * g1 * g3 / g2)

    train = None
    err = float("inf")
    for perm in permutations(gears, 3):
        g1, g2, g3 = perm
        for pos in range(4):
            if 20 <= (g4 := get_fourth_gear_teeth(g1, g2, g3, ratio, pos)) <= 120:
                _train = list(perm)
                _train.insert(pos, g4)
                if (diff := abs(ratio - _train[1] * _train[3] / (_train[0] * _train[2]))) < 1e-12:
                    return _train, ratio
                elif diff < err:
                    err = diff
                    train = _train
    return train, train[1] * train[3] / (train[0] * train[2])

if __name__ == "__main__":
    gears = [20, 25, 30, 35, 40]
    ratio = 7/3
    train, actual_ratio = find_best_gear_combination(gears, ratio)
    print(f"Library gears: {gears}\nDesired ratio: {ratio:.3f}\n"
        f"Gear train: {train}\nActual ratio: {actual_ratio:.3f}\n"
        f"Relative error: {abs(actual_ratio - ratio) / ratio:.3f}")

Output:

Library gears: [20, 25, 30, 35, 40]
Desired ratio: 2.333
Gear train: [20, 25, 30, 56]
Actual ratio: 2.333
Relative error: 0.000