
    Design an algorithm for a learning management system that analyzes a sequence of student assessment scores to find the most significant period of learning progress, where progress is defined as a phase of improving scores followed by a decline due to increasing difficulty levels. The algorithm should return the maximum sum of scores from this identified learning period to help determine optimal points for study material adjustments.
    

    
def max_sum(arr):
    """
    Design an algorithm for a learning management system that analyzes a sequence of student assessment scores to find the most significant period of learning progress, where progress is defined as a phase of improving scores followed by a decline due to increasing difficulty levels. The algorithm should return the maximum sum of scores from this identified learning period to help determine optimal points for study material adjustments.
    

    """
def max_sum(arr):
    """
    Design an algorithm for a learning management system that analyzes a sequence of student assessment scores to find the most significant period of learning progress, where progress is defined as a phase of improving scores followed by a decline due to increasing difficulty levels. The algorithm should return the maximum sum of scores from this identified learning period to help determine optimal points for study material adjustments.
    

    """
    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