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bool isValid(const string& s, int l, int r) {
    unordered_map<char, int> freq;
    for (int i = l; i <= r; i++) {
        freq[s[i]]++;
    }
    int targetFreq = freq[s[l]];
    for (auto& entry : freq) {
        if (entry.second != targetFreq) {
            return false;
        }
    }
    return true;
}

int minimum_k(string s) {
    int n = s.length();
    vector<vector<int>> dp(n, vector<int>(n, INT_MAX)); 
    
    for (int i = 0; i < n; i++) {
        dp[i][i] = 1;
    }
    
    for (int len = 2; len <= n; len++) {
        for (int i = 0; i <= n - len; i++) {
            int j = i + len - 1;
            if (isValid(s, i, j)) {
                dp[i][j] = 1;
                continue;
            }
            for (int k = i; k < j; k++) {
                dp[i][j] = min(dp[i][j], dp[i][k] + dp[k + 1][j]);
            }
        }
    }
    return dp[0][n - 1];
}