Circuit satisfiability problem: Given a Boolean combinatorial circuit composed of AND, OR, and NOT gates, is it satisfiable? A one-output Boolean combinatorial circuit is satisfiable means for the given inputs the output will be $1$.

1. The circuit satisfiability problem belongs to class NP
2. The circuit satisfiability problem is at least as hard as any language in NP
3. The circuit satisfiability is NP-Complete
4. The size of the circuit is $\theta(K^2+1)$
5. If $P \ne NP$, this situation would contradict the NP-Completeness of the problem.

Choose the correct answer from the options given below:

Statement $1$:

Circuit satisfiability belongs to NP because if an input assignment is given, we can verify in polynomial time whether the circuit output is $1$.

So, statement $1$ is true.

Statement $2$:

Circuit satisfiability is at least as hard as any problem in NP.

So, it is NP-hard.

Therefore, statement $2$ is true.

Statement $3$:

A problem is NP-Complete if it is both in NP and NP-hard.

Circuit satisfiability is a standard NP-Complete problem.

So, statement $3$ is true.

Statement $4$:

The size $\theta(K^2+1)$ is not a general property of the circuit satisfiability problem.

So, statement $4$ is false.

Statement $5$:

This statement is not a basic defining property needed here for Circuit SAT classification.

So, statement $5$ is not selected.

Therefore, correct statements are:

$1,2,3$