
    Given a patient's historical readmission risk scores and the number of prediction cycles needed, determine the highest cumulative risk period by analyzing a pattern of consecutive risk scores that repeats across multiple prediction cycles. The risk pattern is formed by repeating the original sequence of risk scores k times to model recurring health conditions, and the goal is to identify the maximum sum of consecutive risk scores in this extended pattern.
    

    
def max_sub_array_sum_repeated(a, n, k):
    """
    Given a patient's historical readmission risk scores and the number of prediction cycles needed, determine the highest cumulative risk period by analyzing a pattern of consecutive risk scores that repeats across multiple prediction cycles. The risk pattern is formed by repeating the original sequence of risk scores k times to model recurring health conditions, and the goal is to identify the maximum sum of consecutive risk scores in this extended pattern.
    

    """
def max_sub_array_sum_repeated(a, n, k):
    """
    Given a patient's historical readmission risk scores and the number of prediction cycles needed, determine the highest cumulative risk period by analyzing a pattern of consecutive risk scores that repeats across multiple prediction cycles. The risk pattern is formed by repeating the original sequence of risk scores k times to model recurring health conditions, and the goal is to identify the maximum sum of consecutive risk scores in this extended pattern.
    

    """
    max_so_far = -2147483648
    max_ending_here = 0
    for i in range(n * k):
        max_ending_here = max_ending_here + a[i % n]
        if max_so_far < max_ending_here:
            max_so_far = max_ending_here
        if max_ending_here < 0:
            max_ending_here = 0
    return max_so_far