<p>You are given an integer array <code>coins</code> of length <code>n</code> which represents the <code>n</code> coins that you own. The value of the <code>i<sup>th</sup></code> coin is <code>coins[i]</code>. You can <strong>make</strong> some value <code>x</code> if you can choose some of your <code>n</code> coins such that their values sum up to <code>x</code>.</p>

<p>Return the <em>maximum number of consecutive integer values that you <strong>can</strong> <strong>make</strong> with your coins <strong>starting</strong> from and <strong>including</strong> </em><code>0</code>.</p>

<p>Note that you may have multiple coins of the same value.</p>

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

<pre>
<strong>Input:</strong> coins = [1,3]
<strong>Output:</strong> 2
<strong>Explanation: </strong>You can make the following values:
- 0: take []
- 1: take [1]
You can make 2 consecutive integer values starting from 0.</pre>

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

<pre>
<strong>Input:</strong> coins = [1,1,1,4]
<strong>Output:</strong> 8
<strong>Explanation: </strong>You can make the following values:
- 0: take []
- 1: take [1]
- 2: take [1,1]
- 3: take [1,1,1]
- 4: take [4]
- 5: take [4,1]
- 6: take [4,1,1]
- 7: take [4,1,1,1]
You can make 8 consecutive integer values starting from 0.</pre>

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

<pre>
<strong>Input:</strong> coins = [1,4,10,3,1]
<strong>Output:</strong> 20</pre>

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

<ul>
	<li><code>coins.length == n</code></li>
	<li><code>1 &lt;= n &lt;= 4 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= coins[i] &lt;= 4 * 10<sup>4</sup></code></li>
</ul>

class Solution:
    def getMaximumConsecutive(self, coins: List[int]) -> int:
        """Solution 1:"""
        # assume coins are sorted
        # say after completing index - 1, we know all values x can be created for 0 <= x <= S_1 where
        # we define S_1 = sum(arr[:i])
        # now, we go to index i
        # it is immediate that we can use arr[i] to construct values arr[i], arr[i] + 1 ... arr[i] + S_1
        # the question is just whether this range includes S_1 + 1
        # if it does, then we know after completing index i that all values x can be created
        # for 0 <= x <= S_2 where S_2 = sum(arr[:i+1])
        # then we keep going
        # but if it does not include S_1 + 1, then we are done (because all values arr[k] for k > i
        # cannot help us to get S_1 + 1 either)
        # note that we include S_1 + 1 if and only if arr[i] <= S_1 + 1
        coins.sort()
        running_sum = 0
        for i in range(len(coins)):
            c = coins[i]
            if c > running_sum + 1:
                return running_sum + 1
            running_sum += c
        return running_sum + 1        

        """Solution 2:"""
    #     possible = True
    #     target = 1
    #     while possible:
    #         if not self.f(target, coins, len(coins) - 1, {}):
    #             possible = False
    #         else:
    #             target += 1
    #     return target
    
    # def f(self, target, coins, idx, memo):
    #     if idx < 0:
    #         return False
    #     if (target, idx) in memo:
    #         return memo[(target, idx)]
        
    #     res = False
    #     if coins[idx] == target:
    #         res = True
    #     elif self.f(target, coins, idx - 1, memo):
    #         res = True
    #     elif coins[idx] < target:
    #         res = self.f(target - coins[idx], coins, idx - 1, memo)
    #     memo[(target, idx)] = res
    #     return res
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