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#include <iostream>
#include <vector>
#include <algorithm>

class Solution {
public:
    std::vector<bool> subsequenceSumAfterCapping(std::vector<int>& nums, int k) {
        int n = nums.size();
        std::sort(nums.begin(), nums.end());
        std::vector<bool> res(n, false);
        std::vector<int> dp(k + 1, 0);
        dp[0] = 1;
        
        int ptr = 0;
        for (int x = 1; x <= n; ++x) {
            while (ptr < n && nums[ptr] <= x) {
                for (int j = k; j >= nums[ptr]; --j) {
                    dp[j] |= dp[j - nums[ptr]];
                }
                ++ptr;
            }
            
            int m = n - ptr;
            for (int j = 0; j <= m; ++j) {
                if (j * x > k) {
                    break;
                }
                if (dp[k - j * x]) {
                    res[x - 1] = true;
                    break;
                }
            }
        }
        
        return res;
    }
};