A zero-indexed array A consisting of N integers is given. An equilibrium index of this array is any integer P such that 0 ≤ P < N and the sum of elements of lower indices is equal to the sum of elements of higher indices, i.e.

A[0] + A[1] + ... + A[P−1] = A[P+1] + ... + A[N−2] + A[N−1].

Sum of zero elements is assumed to be equal to 0. This can happen if P = 0 or if P = N−1.

For example, consider the following array A consisting of N = 7 elements:

A[0] = -7 A[1] = 1 A[2] = 5

A[3] = 2 A[4] = -4 A[5] = 3

A[6] = 0

P = 3 is an equilibrium index of this array, because:

A[0] + A[1] + A[2] = A[4] + A[5] + A[6]

P = 6 is also an equilibrium index, because:

A[0] + A[1] + A[2] + A[3] + A[4] + A[5] = 0

and there are no elements with indices greater than 6.

P = 7 is not an equilibrium index, because it does not fulfill the condition 0 ≤ P < N.

Write a function

def solution(A)

that, given a zero-indexed array A consisting of N integers, returns any of its equilibrium indices. The function should return −1 if no equilibrium index exists.

Assume that:

N is an integer within the range [0..100,000];

each element of array A is an integer within the range [−2,147,483,648..2,147,483,647].

For example, given array A such that

A[0] = -7 A[1] = 1 A[2] = 5

A[3] = 2 A[4] = -4 A[5] = 3

A[6] = 0

the function may return 3 or 6, as explained above.

Complexity:

expected worst-case time complexity is O(N);

expected worst-case space complexity is O(N), beyond input storage (not counting the storage required for input arguments).