Buggy python submission (verdict: Time Limit Exceeded):
N = int(input())
A_list = [int(elem) for elem in input().split(' ')]
assert N >= 4
assert len(A_list) == N

# cummulative sum
cuml_A_list = [0] * N
cuml_A_list[0] = A_list[0]
for i in range(1, N):
    cuml_A_list[i] = cuml_A_list[i-1] + A_list[i]
 
# initialize
best_diff = max(cuml_A_list[0], cuml_A_list[1]-cuml_A_list[0], cuml_A_list[2]-cuml_A_list[1], cuml_A_list[-1]-cuml_A_list[2])\
        - min(cuml_A_list[0], cuml_A_list[1]-cuml_A_list[0], cuml_A_list[2]-cuml_A_list[1], cuml_A_list[-1]-cuml_A_list[2])

left_sep = 0
right_sep = 2 + 1

# time complexity: O(N)?
for mid_sep in range(2, N - 1):
    left_sum = cuml_A_list[mid_sep - 1]
    right_sum = cuml_A_list[-1] - cuml_A_list[mid_sep - 1]
    for left_sep_cand in range(left_sep, mid_sep):
        # obtain the best first separation position
        if abs(cuml_A_list[mid_sep] - 2*cuml_A_list[left_sep_cand]) < abs(cuml_A_list[mid_sep] - 2*cuml_A_list[left_sep]):
            left_sep = left_sep_cand
    for right_sep_cand in range(right_sep, N):
        # obtain the best third separation position
        if abs(cuml_A_list[-1] - cuml_A_list[right_sep_cand] - (cuml_A_list[right_sep_cand] - cuml_A_list[mid_sep]))\
            < abs(cuml_A_list[-1] - cuml_A_list[right_sep] - (cuml_A_list[right_sep] - cuml_A_list[mid_sep])):
            right_sep = right_sep_cand
    diff = max(cuml_A_list[left_sep], cuml_A_list[mid_sep] - cuml_A_list[left_sep], cuml_A_list[right_sep] - cuml_A_list[mid_sep], cuml_A_list[-1] - cuml_A_list[right_sep])\
        - min(cuml_A_list[left_sep], cuml_A_list[mid_sep] - cuml_A_list[left_sep], cuml_A_list[right_sep] - cuml_A_list[mid_sep], cuml_A_list[-1] - cuml_A_list[right_sep])
    best_diff = min(best_diff, diff)
print(best_diff)


Fixed submission:
N = int(input())
A_list = [int(elem) for elem in input().split(' ')]
assert N >= 4
assert len(A_list) == N

# cummulative sum
cuml_A_list = [0] * N
cuml_A_list[0] = A_list[0]
for i in range(1, N):
    cuml_A_list[i] = cuml_A_list[i-1] + A_list[i]

best_diff = max(cuml_A_list[0], cuml_A_list[1]-cuml_A_list[0], cuml_A_list[2]-cuml_A_list[1], cuml_A_list[-1]-cuml_A_list[2])\
        - min(cuml_A_list[0], cuml_A_list[1]-cuml_A_list[0], cuml_A_list[2]-cuml_A_list[1], cuml_A_list[-1]-cuml_A_list[2])

left_sep = 0
right_sep = 2

# time complexity: O(N)
for mid_sep in range(1, N - 2):
    left_sum = cuml_A_list[mid_sep - 1]
    right_sum = cuml_A_list[-1] - cuml_A_list[mid_sep - 1]

    while abs(cuml_A_list[mid_sep] - 2*cuml_A_list[left_sep]) > abs(cuml_A_list[mid_sep] - 2*cuml_A_list[left_sep + 1]):
        left_sep += 1
    while abs(cuml_A_list[-1] - 2*cuml_A_list[right_sep] + cuml_A_list[mid_sep]) > abs(cuml_A_list[-1] - 2*cuml_A_list[right_sep + 1] + cuml_A_list[mid_sep]):
        right_sep += 1
    diff = max(cuml_A_list[left_sep], cuml_A_list[mid_sep] - cuml_A_list[left_sep], cuml_A_list[right_sep] - cuml_A_list[mid_sep], cuml_A_list[-1] - cuml_A_list[right_sep])\
        - min(cuml_A_list[left_sep], cuml_A_list[mid_sep] - cuml_A_list[left_sep], cuml_A_list[right_sep] - cuml_A_list[mid_sep], cuml_A_list[-1] - cuml_A_list[right_sep])
    best_diff = min(best_diff, diff)
print(best_diff)

Buggy python submission (verdict: Time Limit Exceeded):
N = int(input())
A_list = [int(elem) for elem in input().split(' ')]
assert N >= 4
assert len(A_list) == N

# cummulative sum
cuml_A_list = [0] * N
cuml_A_list[0] = A_list[0]
for i in range(1, N):
    cuml_A_list[i] = cuml_A_list[i-1] + A_list[i]
 
# initialize
best_diff = max(cuml_A_list[0], cuml_A_list[1]-cuml_A_list[0], cuml_A_list[2]-cuml_A_list[1], cuml_A_list[-1]-cuml_A_list[2])\
        - min(cuml_A_list[0], cuml_A_list[1]-cuml_A_list[0], cuml_A_list[2]-cuml_A_list[1], cuml_A_list[-1]-cuml_A_list[2])

for mid_sep in range(2, N - 1):
    left_sum = cuml_A_list[mid_sep - 1]
    right_sum = cuml_A_list[-1] - cuml_A_list[mid_sep - 1]
    left_sep = 0
    right_sep = mid_sep + 1
    for left_sep_cand in range(0, mid_sep):
        # obtain the best first separation position
        if abs(cuml_A_list[mid_sep] - 2*cuml_A_list[left_sep_cand]) < abs(cuml_A_list[mid_sep] - 2*cuml_A_list[left_sep]):
            left_sep = left_sep_cand
    for right_sep_cand in range(mid_sep, N):
        # obtain the best third separation position
        if abs(cuml_A_list[-1] - cuml_A_list[right_sep_cand] - (cuml_A_list[right_sep_cand] - cuml_A_list[mid_sep]))\
            < abs(cuml_A_list[-1] - cuml_A_list[right_sep] - (cuml_A_list[right_sep] - cuml_A_list[mid_sep])):
            right_sep = right_sep_cand
    diff = max(cuml_A_list[left_sep], cuml_A_list[mid_sep] - cuml_A_list[left_sep], cuml_A_list[right_sep] - cuml_A_list[mid_sep], cuml_A_list[-1] - cuml_A_list[right_sep])\
        - min(cuml_A_list[left_sep], cuml_A_list[mid_sep] - cuml_A_list[left_sep], cuml_A_list[right_sep] - cuml_A_list[mid_sep], cuml_A_list[-1] - cuml_A_list[right_sep])
    best_diff = min(best_diff, diff)
print(best_diff)


Fixed submission:
N = int(input())
A_list = [int(elem) for elem in input().split(' ')]
assert N >= 4
assert len(A_list) == N

# cummulative sum
cuml_A_list = [0] * N
cuml_A_list[0] = A_list[0]
for i in range(1, N):
    cuml_A_list[i] = cuml_A_list[i-1] + A_list[i]

best_diff = max(cuml_A_list[0], cuml_A_list[1]-cuml_A_list[0], cuml_A_list[2]-cuml_A_list[1], cuml_A_list[-1]-cuml_A_list[2])\
        - min(cuml_A_list[0], cuml_A_list[1]-cuml_A_list[0], cuml_A_list[2]-cuml_A_list[1], cuml_A_list[-1]-cuml_A_list[2])

left_sep = 0
right_sep = 2

# time complexity: O(N)
for mid_sep in range(1, N - 2):
    left_sum = cuml_A_list[mid_sep - 1]
    right_sum = cuml_A_list[-1] - cuml_A_list[mid_sep - 1]

    while abs(cuml_A_list[mid_sep] - 2*cuml_A_list[left_sep]) > abs(cuml_A_list[mid_sep] - 2*cuml_A_list[left_sep + 1]):
        left_sep += 1
    while abs(cuml_A_list[-1] - 2*cuml_A_list[right_sep] + cuml_A_list[mid_sep]) > abs(cuml_A_list[-1] - 2*cuml_A_list[right_sep + 1] + cuml_A_list[mid_sep]):
        right_sep += 1
    diff = max(cuml_A_list[left_sep], cuml_A_list[mid_sep] - cuml_A_list[left_sep], cuml_A_list[right_sep] - cuml_A_list[mid_sep], cuml_A_list[-1] - cuml_A_list[right_sep])\
        - min(cuml_A_list[left_sep], cuml_A_list[mid_sep] - cuml_A_list[left_sep], cuml_A_list[right_sep] - cuml_A_list[mid_sep], cuml_A_list[-1] - cuml_A_list[right_sep])
    best_diff = min(best_diff, diff)
print(best_diff)