The problem can be solved by maintaining a flag whether Takahashi is logged in or not, while scanning the operations.

The sample code is as follows.

#include <iostream>
#include <vector>
#include <string>
using namespace std;

int main() {
    int N;
    cin >> N;

    // A flag that maintains whether he is currently logged in.  Initially he is not logged in
    bool logged_in = false;
    // The number of authorization errors
    int unauthorized = 0;

    for (int i = 0; i < N; ++i) {
        string s;
        cin >> s;

        // If it is a log-in operation, set the flag
        if (s == "login") {
            logged_in = true;
        }
        // If it is a log-out operation, unset the flag
        if (s == "logout") {
            logged_in = false;
        }
        // If it is an attempt to access a private page while not logged in, increment the number of authorization errors
        if (s == "private" && !logged_in) {
            unauthorized++;
        }
    }

    cout << unauthorized << endl;
    
    return 0;
}

N = int(input())

# A flag that maintains whether he is currently logged in.  Initially he is not logged in
logged_in = False
# The number of authorization errors
unauthorized = 0

for i in range(N):
    s = input()
    # If it is a log-in operation, set the flag
    if s == 'login':
        logged_in = True
    # If it is a log-out operation, unset the flag
    if s == 'logout':
        logged_in = False
    # If it is an attempt to access a private page while not logged in, increment the number of authorization errors
    if s == 'private' and not logged_in:
        unauthorized += 1

print(unauthorized)