onsider a sequence A $=($a$1,$ a$2,...,$ an$)$ consisting of distinct positive

integers. Let each element ai of A$,$ where i $=2$k for some positive integer k$,$ be positioned in the

sorted list of A such that it is either $($i$)$ at its sorted position or $($ii$)$ within two positions $($either to

the left or right$)$ from its sorted position.

For example, sequence A :$=($a$\_1,...,$ a$\_10),$ the possible positions of the element a$\_2$ in the sorted list of A can be $1,2,3$ and $4.$

$(1)$ Derive a formula that calculates the total number of possible sequences for any given n$.$

$(2)$ Prove the correctness of your formula by mathematical induction. Your answer should

include the base case $($n $=1),$ hypothesis, and inductive stepProblem $1.(25$ points$)$ Consider a sequence $A=(a_1,a_2,$dots,$a_n)$ consisting of distinct positive

integers. Let each element $a_i$ of $A,$ where $i=2^k$ for some positive integer $k,$ be positioned in the

sorted list of A such that it is either $($i$)$ at its sorted position or $($ii$)$ within two positions $($either to

the left or right$)$ from its sorted position.

For example, given a sequence $A$:$=(a_1,$dots,$a_{10}),$ the possible positions of the element $a_2$ in the

sorted list of A can be $1,2,3,4.$

$(1)$ Derive a formula that calculates the total number of possible sequences for any given $n.$

$(2)$ Prove the correctness of your formula by mathematical induction. Your answer should

include the base case $(n=1),$ hypothesis, and inductive step.