There are n persons numbered from 0 to n - 1 and a door. Each person can enter or exit through the door once, taking one second.

You are given a non-decreasing integer array arrival of size n, where arrival[i] is the arrival time of the ith person at the door. You are also given an array state of size n, where state[i] is 0 if person i wants to enter through the door or 1 if they want to exit through the door.

If two or more persons want to use the door at the same time, they follow the following rules:

• If the door was not used in the previous second, then the person who wants to exit goes first.
• If the door was used in the previous second for entering, the person who wants to enter goes first.
• If the door was used in the previous second for exiting, the person who wants to exit goes first.
• If multiple persons want to go in the same direction, the person with the smallest index goes first.

Return an array answer of size n where answer[i] is the second at which the ith person crosses the door.

Note that:

• Only one person can cross the door at each second.
• A person may arrive at the door and wait without entering or exiting to follow the mentioned rules.

 

Example 1:

Input: arrival = [0,1,1,2,4], state = [0,1,0,0,1]
Output: [0,3,1,2,4]
Explanation: At each second we have the following:
- At t = 0: Person 0 is the only one who wants to enter, so they just enter through the door.
- At t = 1: Person 1 wants to exit, and person 2 wants to enter. Since the door was used the previous second for entering, person 2 enters.
- At t = 2: Person 1 still wants to exit, and person 3 wants to enter. Since the door was used the previous second for entering, person 3 enters.
- At t = 3: Person 1 is the only one who wants to exit, so they just exit through the door.
- At t = 4: Person 4 is the only one who wants to exit, so they just exit through the door.

Example 2:

Input: arrival = [0,0,0], state = [1,0,1]
Output: [0,2,1]
Explanation: At each second we have the following:
- At t = 0: Person 1 wants to enter while persons 0 and 2 want to exit. Since the door was not used in the previous second, the persons who want to exit get to go first. Since person 0 has a smaller index, they exit first.
- At t = 1: Person 1 wants to enter, and person 2 wants to exit. Since the door was used in the previous second for exiting, person 2 exits.
- At t = 2: Person 1 is the only one who wants to enter, so they just enter through the door.

 

Constraints:

• n == arrival.length == state.length
• 1 <= n <= 105
• 0 <= arrival[i] <= n
• arrival is sorted in non-decreasing order.
• state[i] is either 0 or 1.