Testing primality for large numbers

The number \(121\) is composite.
The number \(1211\) is composite.
The number \(12111\) is composite.
The number \(121111\) is composite.

Let \(n\) be the smallest positive integer such that \(12\underbrace{11...11}_{n \text{ times}}\) is prime.

Find \(n\).

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