A long string of lies

By assumption, we see that if statement $1$ is $true$ , then it follows that statement $2$ is $false$ , and statements $3$ and $4$ become automatically $true$ . The next four statements will follow suit.

Knowing that, we can determine which of the statements are equivalent with the last statement.

That is,

$3(3^{20! -1 } -1) = 3^{20!} - 3$ .

$3^{20!} \equiv 1 (mod$ $4)$ , meaning that whether statement $3^{20!}$ is true or false is the same as statement 1. That is if $3^{20!}$ exists, but it does not.

It then follows that statement $3^{20!} - 3$ has the same nature with statement 2. Then it follows as well that statement $3^{20!} - 3 - 3^{5}$ follows statement $(2- 3) = -1$ or statement $3$ . Since $3^{20!} - 3 - 3^{5}$ is a lie, it then follows that statements 1 and 4 are lies, and statement 2 is true.

Then, we get that $7^{12} \equiv 1 (mod$ $4)$ . Thus, statement $7^{12} + 1$ follows statement $2$ , which is $\boxed {true}$ .