
    Given a patient's historical readmission risk scores from previous hospital visits, determine the highest cumulative risk pattern by analyzing sequences of increasing risk scores up to their current hospital visit, while also considering a specific future predicted hospital visit. The analysis should identify the sequence of hospital visits that shows an escalating pattern of readmission risk, ending with the current visit, plus the inclusion of one specific future visit's predicted risk score. The sequence must maintain a strictly increasing risk score pattern to accurately reflect deteriorating patient conditions. The goal is to identify the maximum sum of risk scores that could indicate the most concerning trajectory of patient health, helping to prioritize preventive interventions.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Given a patient's historical readmission risk scores from previous hospital visits, determine the highest cumulative risk pattern by analyzing sequences of increasing risk scores up to their current hospital visit, while also considering a specific future predicted hospital visit. The analysis should identify the sequence of hospital visits that shows an escalating pattern of readmission risk, ending with the current visit, plus the inclusion of one specific future visit's predicted risk score. The sequence must maintain a strictly increasing risk score pattern to accurately reflect deteriorating patient conditions. The goal is to identify the maximum sum of risk scores that could indicate the most concerning trajectory of patient health, helping to prioritize preventive interventions.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Given a patient's historical readmission risk scores from previous hospital visits, determine the highest cumulative risk pattern by analyzing sequences of increasing risk scores up to their current hospital visit, while also considering a specific future predicted hospital visit. The analysis should identify the sequence of hospital visits that shows an escalating pattern of readmission risk, ending with the current visit, plus the inclusion of one specific future visit's predicted risk score. The sequence must maintain a strictly increasing risk score pattern to accurately reflect deteriorating patient conditions. The goal is to identify the maximum sum of risk scores that could indicate the most concerning trajectory of patient health, helping to prioritize preventive interventions.
    

    """
    dp = [[0 for i in range(n)] for i in range(n)]
    for i in range(n):
        if a[i] > a[0]:
            dp[0][i] = a[i] + a[0]
        else:
            dp[0][i] = a[i]
    for i in range(1, n):
        for j in range(n):
            if a[j] > a[i] and j > i:
                if dp[i - 1][i] + a[j] > dp[i - 1][j]:
                    dp[i][j] = dp[i - 1][i] + a[j]
                else:
                    dp[i][j] = dp[i - 1][j]
            else:
                dp[i][j] = dp[i - 1][j]
    return dp[index][k]