I stared at this for a bit before realizing it's a 4D DP problem.
Model the problem as a dynamic programming over the array indices, where the state tracks the remaining counts of length-1, length-2, and length-3 subarrays. At each index, consider skipping the element or taking a subarray of length 1, 2, or 3 (if within limits and bounds), and maximize the sum. Optimize by using memoization or iterative DP with state compression.
Pro tip: Clarify that subarrays must be non-overlapping and contiguous, and that limits are per-length, not total. Mention that greedy approaches fail because local choices affect future availability, so DP is necessary.
Confirm array size, possible negative values, and that limits are per-length. Discuss edge cases like empty array, limits zero, or all negatives.
Define dp[i][a][b][c] as max sum from index i with a,b,c remaining picks for lengths 1,2,3. Recurrence: dp[i][a][b][c] = max(skip, take1, take2, take3) where takes are valid if counts >0 and i+len <= n.
Note that a,b,c are bounded by x,y,z (≤ n). Use memoization or iterative DP with rolling array over i. Complexity O(n * x * y * z).
If all values negative, optimal may be to take nothing (sum 0) if allowed, or must take? Clarify. Base case: dp[n][*][*][*] = 0. Ensure skip option always available.
Walk through a small example. Mention that if limits are large, DP may be heavy; consider alternative if limits are small or array is short.
AI-generated suggestions, not part of the candidate's original notes. May be inaccurate — verify before relying on them.