
    Design an algorithm for a personalized learning system that analyzes a sequence of student performance scores to identify the most effective learning path, where scores initially show improvement as concepts build upon each other, followed by a strategic decrease in difficulty to consolidate learning. The goal is to find the learning path with the maximum cumulative score that follows this pattern of progressive improvement followed by strategic difficulty adjustment, helping optimize the student's mastery of connected topics.
    

    
def max_sum(arr):
    """
    Design an algorithm for a personalized learning system that analyzes a sequence of student performance scores to identify the most effective learning path, where scores initially show improvement as concepts build upon each other, followed by a strategic decrease in difficulty to consolidate learning. The goal is to find the learning path with the maximum cumulative score that follows this pattern of progressive improvement followed by strategic difficulty adjustment, helping optimize the student's mastery of connected topics.
    

    """
def max_sum(arr):
    """
    Design an algorithm for a personalized learning system that analyzes a sequence of student performance scores to identify the most effective learning path, where scores initially show improvement as concepts build upon each other, followed by a strategic decrease in difficulty to consolidate learning. The goal is to find the learning path with the maximum cumulative score that follows this pattern of progressive improvement followed by strategic difficulty adjustment, helping optimize the student's mastery of connected topics.
    

    """
    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