a$)($i$)$ Design and Explaina PDA to accept each of the following language

$($ii$)$ The set of all string with twice as many $0\text{'}$s and $1\text{'}$s$.$

b$)($i$)$ Convert the following CFG to PDA and analyze the answer$($a$+$b$)$ and a$++.$

$Ia|b|Ia|Ib|IO|II|$

$EI|E+E|E**E|(E)|$

$($ii$)$ Convert the following CFG to PDA by empty stack.

$S0S\frac{1}{A}$;

$A1A\frac{0}{S}\frac{?}{}$ Infer whether $0101$ belongs to N$($M$).$