Problem $5.\backslash ((10+10=20\backslash )$ points$)$ Read Section $10\mathrm{.}6$ carefully before attempting this problem.

Analyze the running time of the following algorithm using a step count analysis as shown in the Horner scheme $($Example $10\mathrm{.}40).$

$```$

$//$ search a key in an array a$[1..$n$]$ of length n

search$($a$,$ n$,$ key$)$ cost times

for k in $(1..$n$)$ do c$1[]$

if a$[$k$]=$key then c$2[]$

return k c$3[]$

endfor c$4[]$

return false c$5[]$

$```$

$($a$)$ Fill in the $[]$s in the above code each with a number or an expression involving n that expresses the step count for the line of code.

$($b$)$ Determine the worst$-$case complexity of this algorithm and give it in the $\backslash (\backslash$Theta $\backslash )$ notation. Show your work and explain using the definition of $\backslash (\backslash$Theta $\backslash )$ involving the inequalities.