<p>Given an integer array <code>nums</code> and an integer <code>k</code>, find three non-overlapping subarrays of length <code>k</code> with maximum sum and return them.</p>

<p>Return the result as a list of indices representing the starting position of each interval (<strong>0-indexed</strong>). If there are multiple answers, return the lexicographically smallest one.</p>

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

<pre>
<strong>Input:</strong> nums = [1,2,1,2,6,7,5,1], k = 2
<strong>Output:</strong> [0,3,5]
<strong>Explanation:</strong> Subarrays [1, 2], [2, 6], [7, 5] correspond to the starting indices [0, 3, 5].
We could have also taken [2, 1], but an answer of [1, 3, 5] would be lexicographically larger.
</pre>

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

<pre>
<strong>Input:</strong> nums = [1,2,1,2,1,2,1,2,1], k = 2
<strong>Output:</strong> [0,2,4]
</pre>

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

<ul>
	<li><code>1 &lt;= nums.length &lt;= 2 * 10<sup>4</sup></code></li>
	<li><code>1 &lt;= nums[i] &lt;&nbsp;2<sup>16</sup></code></li>
	<li><code>1 &lt;= k &lt;= floor(nums.length / 3)</code></li>
</ul>

class Solution:
    def build_range_sum(self, nums: List[int], k: int) -> Dict[int, int]:
        res = {}
        cur_sum = 0
        for i in range(len(nums)-k+1):
            if i==0:
                cur_sum = sum(nums[i:i+k])
            else:
                cur_sum -= nums[i-1]
                cur_sum += nums[i+k-1]
            res[i] = cur_sum
        return res
    
    def dfs(self, idx: int, remain: int, k: int) -> Tuple[int, List[int]]:
        if idx+remain*k > self.n:
            return [self.inf, []]
        elif remain == 0:
            return [0, []]
        elif (idx, remain) in self.cache:
            return self.cache[(idx, remain)]
        else:
            # 1. not select the current idx
            max_sum1, idxs1 = self.dfs(idx+1, remain, k)
            # 2. select current idx
            max_sum2, idxs2 = self.dfs(idx+k, remain-1, k)
            if max_sum2+self.range_sum[idx] >= max_sum1:
                res = [max_sum2+self.range_sum[idx], [idx]+idxs2[:]]
            else:
                res = [max_sum1, idxs1[:]]
            self.cache[(idx, remain)] = res
            return res
        
    
    def maxSumOfThreeSubarrays(self, nums: List[int], k: int) -> List[int]:
        self.n = len(nums)
        self.cache = {}
        self.range_sum = self.build_range_sum(nums, k)
        self.inf = -10**10
        
        max_sum, idxs = self.dfs(0, 3, k)
        return idxs
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