Sure! Let's create a test case generator for the provided problem. This generator will create random test cases while ensuring that the constraints are respected.

Here is the Python code for the test case generator:

```python
import random
import string

def tcgen():
    # Generate a random value for N within the specified range
    N = random.randint(1, 2 * 10**5)
    
    # Generate N random strings, each string's length between 1 and 10,
    # and consisting of lowercase English letters.
    strings = []
    for _ in range(N):
        length = random.randint(1, 10)
        s = ''.join(random.choices(string.ascii_lowercase, k=length))
        strings.append(s)
    
    # Return N and the list of strings
    return N, strings

# Example usage:
N, strings = tcgen()
print(N)
for s in strings:
    print(s)
```

### Explanation
- **N Generation**: It generates a random integer \(N\) in the range \([1, 2 \times 10^5]\).
- **String List Generation**: It generates \(N\) random strings. Each string's length is between 1 and 10, and the string consists of lowercase English letters.

### Example Output
When you run the `tcgen()` function, it might produce an output like the following:

```
5
abc
defgh
ijklmnop
qrstuvwx
yz
```

This output represents:
- \(N = 5\) (number of draws)
- 5 strings: "abc", "defgh", "ijklmnop", "qrstuvwx", "yz"

You can use this generator to create random test cases for testing the solution to the problem.