
    Given a patient's daily medication adherence scores, determine the longest period of consistent behavior where the compliance first improved and then declined, and calculate the total adherence score for this period to identify significant patterns in medication-taking behavior. The adherence scores are represented as positive numbers where higher values indicate better compliance, and the goal is to find the maximum sum of scores during any period that shows this improvement-then-decline pattern.
    

    
def max_sum(arr):
    """
    Given a patient's daily medication adherence scores, determine the longest period of consistent behavior where the compliance first improved and then declined, and calculate the total adherence score for this period to identify significant patterns in medication-taking behavior. The adherence scores are represented as positive numbers where higher values indicate better compliance, and the goal is to find the maximum sum of scores during any period that shows this improvement-then-decline pattern.
    

    """
def max_sum(arr):
    """
    Given a patient's daily medication adherence scores, determine the longest period of consistent behavior where the compliance first improved and then declined, and calculate the total adherence score for this period to identify significant patterns in medication-taking behavior. The adherence scores are represented as positive numbers where higher values indicate better compliance, and the goal is to find the maximum sum of scores during any period that shows this improvement-then-decline pattern.
    

    """
    MSIBS = arr[:]
    for i in range(len(arr)):
        for j in range(0, i):
            if arr[i] > arr[j] and MSIBS[i] < MSIBS[j] + arr[i]:
                MSIBS[i] = MSIBS[j] + arr[i]
    MSDBS = arr[:]
    for i in range(1, len(arr) + 1):
        for j in range(1, i):
            if arr[-i] > arr[-j] and MSDBS[-i] < MSDBS[-j] + arr[-i]:
                MSDBS[-i] = MSDBS[-j] + arr[-i]
    max_sum = float('-Inf')
    for (i, j, k) in zip(MSIBS, MSDBS, arr):
        max_sum = max(max_sum, i + j - k)
    return max_sum