Design a push$-$down automaton for each language. All languages are over the alphabet $\backslash$Sigma $=\{0,1\}.$

a$)$ L$1=\{$w in $\backslash$Sigma $|$ the middle symbol of w is a $1\}$

b$)$ L$2=\{$w in $\backslash$Sigma $|$w $=0^$n$1^$m where n$,$m $>=0$ and n $<$ m$\}$

c$)$ L$3=\{$w in $\backslash$Sigma $|$w $=0^$n$1^$m where n$,$m $>=0$ and n $=$ m$\}$

d$)$ L$4=\{$w in $\backslash$Sigma $|$w $=0^$n$1^$n$0^$m$1^$m where n$,$m $>=0\}$

e$)$ L$5=\{$w in $\backslash$Sigma $|$w has more $0$s than $1$s$\}$