If we choose some prime, say $11$, we can concatenate one digit to the left and one to the right to obtain another prime, for example, $2113$, we can do the same with $2113$ and obtain $121139$, a prime, and so on...

Let us call some prime (written in decimal notation as $a_1...a_k$) a *$1$-concatenable* prime if there exist digits $b_1$ and $c_1$ such that $b_1a_1...a_kc_1$ is prime and digits $b_2$ and $c_2$ such that $b_2b_1a_1...a_kc_1c_2$ is prime and...and digits $b_l$ and $c_l$ such that $b_l...b_2b_1a_1...a_kc_1c_2...c_l$ is prime, and so on...

That is, a prime number is *$1$-concatenable* prime if, starting from it, we can build larger and larger primes in such a way that at each new step we concatenate one digit from the left and one from the right, and if we can repeat that process an infinite number of times.

Is there at least one *$1$-concatenable* prime?

If there is none, are there chains of arbitrary length, that is, is it true that for every $n \in \mathbb N$ there exists some prime $p$ such that $p$ is $n$ times concatenable in the described way, that is, by concatenating at each step one digit from the left and one from the right?