
    Design a fraud detection algorithm that analyzes a customer's transaction sequence to find the highest-risk spending pattern, where risk is measured as the sum of transaction amounts that show an increasing-then-decreasing behavior. The pattern of interest starts with progressively larger transactions (possibly testing limits) followed by declining transaction amounts, and the algorithm should identify the sequence with the maximum total value that follows this pattern.
    

    
def max_sum(arr):
    """
    Design a fraud detection algorithm that analyzes a customer's transaction sequence to find the highest-risk spending pattern, where risk is measured as the sum of transaction amounts that show an increasing-then-decreasing behavior. The pattern of interest starts with progressively larger transactions (possibly testing limits) followed by declining transaction amounts, and the algorithm should identify the sequence with the maximum total value that follows this pattern.
    

    """
def max_sum(arr):
    """
    Design a fraud detection algorithm that analyzes a customer's transaction sequence to find the highest-risk spending pattern, where risk is measured as the sum of transaction amounts that show an increasing-then-decreasing behavior. The pattern of interest starts with progressively larger transactions (possibly testing limits) followed by declining transaction amounts, and the algorithm should identify the sequence with the maximum total value that follows this 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