$(4$ points$)$ Given the following pseudocode, solve the following problems:

INITIALIZE x TO O

FOR i FROM $1$ TO N DO

PRINT x

IF x $>0$ THEN

PRINT x

FOR j FROM $1$ TO $\}\backslash$sqrt$\{\}\{$N$\}\backslash$mathrm$\{$ DO

PRINT j

INCREMENT j BY $1$

END FOR

END IF

FOR k FROM $2$ TO N DO

PRINT k

INCREMENT k BY k $*2$

END FOR

INCREMENT i BY i $*3$

INCREMENT x BY $1$

END FOR

$($a$)(3$ points$)$ Compute the exact number of steps required by the algorithm and provide a

proof that your resulting complexity function is correct. To receive full credit, organize

your work, and include the solutions to all relevant summations.

$($b$)(1$ point$)$ Determine the Big$-$O time complexity of the function you derived in part $($a$).$

Explain in detail how you arrived at this conclusion.