Given an integer array banned and two integers n and max_sum, determine the maximum number of integers you can choose while following the below rules:

1. The selected integers must fall within the range $[1, n]$.

2. Each integer can be chosen at most once.

3. No selected integer can be present in the banned array.

4. The sum of the selected integers must not exceed max_sum.

Your goal is to return the maximum count of integers that can be chosen while satisfying all the above rules.

Constraints:

• $1 \leq$ banned.length $\leq 10^3$

• $1 \leq$ banned[i] $\leq$ n $\leq 10^3$

• $1 \leq$ max_sum $\leq 10^9$

To essence of the solution is to use the sort and search pattern. We first sort the banned array, then take advantage of the sorted nature of the sequence of numbers being evaluated and the sorted banned array.

The approach involves iterating through numbers from $1$ to $n$ while maintaining a banned_idx pointer to track the position in the banned array. This pointer helps determine whether the current number is banned by comparing it with the next value in the banned array. If the number matches the value pointed to by banned_idx, it is skipped, and the pointer is moved to the next position in the banned array. This avoids scanning each number in the array.

As we iterate through the numbers, we add them to the count of valid integers if they are not banned. For each valid number, we subtract it from max_sum. Suppose the subtraction causes max_sum to drop below zero. In that case, we conclude the process and return the current count, representing the maximum number of integers that can be included without exceeding the maximum sum limit.

Now, let’s look at the solution:

1. Sort the banned array.

2. Initialize the variables:

   1. Start with a banned_idx pointer at $0$ to keep track of the current position in the banned array.

   2. Set a counter, count, and count to $0$ to record how many numbers are included.

3. Iterate through each number i —from $1$ to $n$.

   1. Compare the current number i with the value at banned[banned_idx]. If i match banned[banned_idx], the number is banned. Skip this number and move banned_idx forward to the next position. Continue moving banned_idx until the next value is differs from the current one to handle duplicates.

   2. If the number is not banned, subtract its value from max_sum. After subtracting a valid number, stop iterating if max_sum drops below $0$. At this point, the count represents the maximum number of valid numbers that can be included without exceeding max_sum, so we return the count as the final result.

   3. Otherwise, increment the count to indicate that this number has been included and keep iterating.

4. After iterating through each number, return the total count of numbers included.

Let’s look at the following illustration to get a better understanding of the solution: