Your robot buddy GNPT$-4$ has come up with a revolutionary new strategy to prove that it is in fact equal in computational power to its more well$-$known cousin. It has a simple yet brilliant proof strategy: it will start by proving that P in fact equals the set of Turing$-$decidable languages, by showing that every decider runs in polynomial time. Once it has done this, it will obtain as a corollary that NP is also equal to this set, and the result will follow. GNPT$-4$ would like you to check its generated proof, and has generously offered you half of the million dollar bounty for doing so$.$ Unfortunately, you$$re starting to have some concerns about the claim that every decider runs in polynomial time. GNPT$-4$s proof of this claim is $2123$ pages long, so you don$$t really feel like checking it in detail for a flaw. Instead, you have a much better idea: you$$ll provide an explicit counterexample of a machine that does not run in polynomial time.

$1.$ Provide a low level description in Morphett notation of a $(1-$tape deterministic$)$ TM over input alphabet $=\{$a$\}$ that accepts every string, has at most $20$ states, and has time complexity f$($n$)$ such that $2^$n $$ f$($n$)2^(2$n$+1)$ for all n$.$

$2.$ Provide a low level description in Morphett notation of a $(1-$tape deterministic$)$ TM over input alphabet $=\{$a$\}$ that accepts every string, has at most $40$ states, and has time complexity exactly $2^$n $.$