
    Design an algorithm for a patient monitoring system that analyzes vital sign measurements to identify the most significant increasing trend in a patient's condition up to a current checkpoint, while accounting for a future critical measurement point. The algorithm should find the maximum sum of increasing vital sign readings from the start of monitoring up to the current checkpoint, incorporating a specific future measurement that medical staff needs to consider for complication prediction. The vital signs sequence must be strictly increasing to indicate a consistent trend in the patient's condition. The future measurement point must be considered in the final analysis to help predict potential complications.
    

    
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for a patient monitoring system that analyzes vital sign measurements to identify the most significant increasing trend in a patient's condition up to a current checkpoint, while accounting for a future critical measurement point. The algorithm should find the maximum sum of increasing vital sign readings from the start of monitoring up to the current checkpoint, incorporating a specific future measurement that medical staff needs to consider for complication prediction. The vital signs sequence must be strictly increasing to indicate a consistent trend in the patient's condition. The future measurement point must be considered in the final analysis to help predict potential complications.
    

    """
def max_sum_increasing_subseq(a, n, index, k):
    """
    Design an algorithm for a patient monitoring system that analyzes vital sign measurements to identify the most significant increasing trend in a patient's condition up to a current checkpoint, while accounting for a future critical measurement point. The algorithm should find the maximum sum of increasing vital sign readings from the start of monitoring up to the current checkpoint, incorporating a specific future measurement that medical staff needs to consider for complication prediction. The vital signs sequence must be strictly increasing to indicate a consistent trend in the patient's condition. The future measurement point must be considered in the final analysis to help predict potential complications.
    

    """
    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]