<p>You are given an integer array <code>coins</code> representing coins of different denominations and an integer <code>amount</code> representing a total amount of money.</p>

<p>Return <em>the fewest number of coins that you need to make up that amount</em>. If that amount of money cannot be made up by any combination of the coins, return <code>-1</code>.</p>

<p>You may assume that you have an infinite number of each kind of coin.</p>

<p>&nbsp;</p>
<p><strong class="example">Example 1:</strong></p>

<pre>
<strong>Input:</strong> coins = [1,2,5], amount = 11
<strong>Output:</strong> 3
<strong>Explanation:</strong> 11 = 5 + 5 + 1
</pre>

<p><strong class="example">Example 2:</strong></p>

<pre>
<strong>Input:</strong> coins = [2], amount = 3
<strong>Output:</strong> -1
</pre>

<p><strong class="example">Example 3:</strong></p>

<pre>
<strong>Input:</strong> coins = [1], amount = 0
<strong>Output:</strong> 0
</pre>

<p>&nbsp;</p>
<p><strong>Constraints:</strong></p>

<ul>
	<li><code>1 &lt;= coins.length &lt;= 12</code></li>
	<li><code>1 &lt;= coins[i] &lt;= 2<sup>31</sup> - 1</code></li>
	<li><code>0 &lt;= amount &lt;= 10<sup>4</sup></code></li>
</ul>

class Solution:
    def coinChange(self, coins: List[int], amount: int) -> int:
        
        # Sort the coins in descending order
        reversedOrderedCoins = sorted(coins, reverse=True)

        def dp(currAmount: int) -> int:
            if currAmount == amount:
                return 0
            elif currAmount > amount:
                return -1

            if currAmount in memo:
                return memo[currAmount]

            minimum = -1
            for coin in coins:
                candidate = dp(currAmount + coin)
                
                # If candidate reaches the amount, break the loop early
                if candidate == 0:
                    memo[currAmount] = 1
                    return 1
                
                if minimum == -1 and candidate >= 0:  # First valid solution
                    minimum = candidate
                elif candidate != -1:  # Ignore invalid paths
                    minimum = min(minimum, candidate)

            if minimum == -1:
                memo[currAmount] = -1
            else:
                memo[currAmount] = minimum + 1

            return memo[currAmount]

        memo = {}
        return dp(0)

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