<p>A magician has various spells.</p>

<p>You are given an array <code>power</code>, where each element represents the damage of a spell. Multiple spells can have the same damage value.</p>

<p>It is a known fact that if a magician decides to cast a spell with a damage of <code>power[i]</code>, they <strong>cannot</strong> cast any spell with a damage of <code>power[i] - 2</code>, <code>power[i] - 1</code>, <code>power[i] + 1</code>, or <code>power[i] + 2</code>.</p>

<p>Each spell can be cast <strong>only once</strong>.</p>

<p>Return the <strong>maximum</strong> possible <em>total damage</em> that a magician can cast.</p>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">power = [1,1,3,4]</span></p>

<p><strong>Output:</strong> <span class="example-io">6</span></p>

<p><strong>Explanation:</strong></p>

<p>The maximum possible damage of 6 is produced by casting spells 0, 1, 3 with damage 1, 1, 4.</p>
</div>

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

<div class="example-block">
<p><strong>Input:</strong> <span class="example-io">power = [7,1,6,6]</span></p>

<p><strong>Output:</strong> <span class="example-io">13</span></p>

<p><strong>Explanation:</strong></p>

<p>The maximum possible damage of 13 is produced by casting spells 1, 2, 3 with damage 1, 6, 6.</p>
</div>

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

<ul>
	<li><code>1 &lt;= power.length &lt;= 10<sup>5</sup></code></li>
	<li><code>1 &lt;= power[i] &lt;= 10<sup>9</sup></code></li>
</ul>

class Solution:
    def maximumTotalDamage(self, power: List[int]) -> int:
        # 1 2 3 4
        # 2 0 1 1
        # T (T T T) X X T
        # T X X T X X T
        cnt = Counter(power)
        nums = sorted(list(cnt.keys()))
        n = len(nums)
        dp = [-1] * n
        @cache
        def dfs(i):
            
            if i < 0:
                return 0
            if  dp[i] != -1:
                return dp[i]
                        
            # for i in range(1,6):
            #     # 1 ~5
            candicate = [ dfs(i-j)  for j in range(1,6) if i-j>=0 and nums[i]>= nums[i-j]+3  ]
            if not candicate:
                dp[i] = nums[i]*cnt[nums[i]]
                return dp[i]            
            dp[i] = max(candicate) +nums[i]*cnt[nums[i]]            
            return  dp[i]
        # print(f"{dfs(n-1)=}")
        # print(f"{dfs(n-2)=}")
        dfs(n-1)
        dfs(n-2)
        dfs(n-3)
        print(f"{dp=}")
        if len(dp)>=5:
            return max(dp[-5:])
        return max(dp)
        


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