To generate test cases for the problem "Find the number of different values that the median of X_1, X_2, \cdots, X_N can take", we need to account for various cases within the constraints provided. The constraints are:

- \(2 \leq N \leq 2 \times 10^5\)
- \(1 \leq A_i \leq B_i \leq 10^9\)

Here's a Python function to generate test cases for the problem:

```python
import random

def tcgen():
    # Generate a random value for N within the constraints
    N = random.randint(2, 2 * 10**5)
    
    # Generate pairs (A_i, B_i) where 1 <= A_i <= B_i <= 10^9
    AB = []
    for _ in range(N):
        A = random.randint(1, 10**9)
        B = random.randint(A, 10**9)
        AB.append((A, B))
    
    return N, AB

# Example usage:
N, AB = tcgen()
print(N)
for A, B in AB:
    print(A, B)
```

This function generates a random integer \(N\) within the specified range and then creates \(N\) pairs of integers \((A_i, B_i)\) such that \(1 \leq A_i \leq B_i \leq 10^9\). Each pair is generated to ensure \(A_i \leq B_i\).

You can use this function to generate test cases for the problem. Note that the output of the function is in the correct format for input to the competitive programming problem.

### Explanation:
1. **N Generation**: Selects a random value for \(N\) which determines the number of pairs.
2. **Pair Generation**: For each pair, selects \(A_i\) randomly between 1 and \(10^9\). Then, \(B_i\) is selected such that it is greater than or equal to \(A_i\).

This should allow you to test the problem for a wide range of inputs, ensuring robustness and correctness of your solution.