def largest_first(coins, amount):
    if type(amount) is not int or amount < 0 or any(type(coin) is not int or coin < 1 for coin in coins):
        raise ValueError("Positive integer coins and nonnegative amount required")
    chosen = []
    for coin in sorted(coins, reverse=True):
        while coin <= amount:
            amount -= coin
            chosen.append(coin)
    return chosen, amount

chosen, remainder = largest_first([1, 3, 4], 6)
print(chosen, remainder)
print([3, 3], len([3, 3]) < len(chosen))