
    Design a medication safety analysis system that determines the optimal combination of medications from a patient's prescription history that minimizes dangerous drug interactions while including a newly proposed medication. The system must evaluate medications up to a specific provider's prescription point (index) in the patient's history and consider adding a new medication at a later position (k) in the sequence. Given a list of interaction severity scores where higher values indicate safer combinations, find the safest possible medication combination that follows the sequential prescription order and must include the new medication at position k. The combination must maintain an increasing safety profile, meaning each subsequent medication in the sequence must have a higher interaction safety score than the previous ones.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a medication safety analysis system that determines the optimal combination of medications from a patient's prescription history that minimizes dangerous drug interactions while including a newly proposed medication. The system must evaluate medications up to a specific provider's prescription point (index) in the patient's history and consider adding a new medication at a later position (k) in the sequence. Given a list of interaction severity scores where higher values indicate safer combinations, find the safest possible medication combination that follows the sequential prescription order and must include the new medication at position k. The combination must maintain an increasing safety profile, meaning each subsequent medication in the sequence must have a higher interaction safety score than the previous ones.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design a medication safety analysis system that determines the optimal combination of medications from a patient's prescription history that minimizes dangerous drug interactions while including a newly proposed medication. The system must evaluate medications up to a specific provider's prescription point (index) in the patient's history and consider adding a new medication at a later position (k) in the sequence. Given a list of interaction severity scores where higher values indicate safer combinations, find the safest possible medication combination that follows the sequential prescription order and must include the new medication at position k. The combination must maintain an increasing safety profile, meaning each subsequent medication in the sequence must have a higher interaction safety score than the previous ones.
    

    """
    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]