$(1)$ Define a PDA as a $7-$tuple and describe each of the components.

$(2)$ Create a PDA that recognizes the following context free language with terminals $\backslash (\backslash \{\backslash$mathrm$\{$a$\},\backslash$mathrm$\{$b$\}\backslash \}\backslash )\backslash ($ L$=\backslash \{$w $\backslash$mid $\backslash )$ number of a$\text{'}$s $=$ twice the number of b$\text{'}$s; String $\backslash ($ w $\backslash )$ can only have a$\text{'}$s followed by b$\text{'}$s or b$\text{'}$s followed by a$\text{'}$s$\backslash \}$

i$.$e$.,$ it should accept aab, aaaabb, baa, bbaaaa,.. and so on$.$

$($i$)$ Describe your algorithm

$($ii$)$ Give the description as a complete $7-$tuple with a transition diagram

$($iii$)$ Show configuration sequences on aaaabb leading to acceptance

$(3)$ Create a PDA that recognizes the following context free language with terminals $\backslash (\backslash \{$a$,$ b$,$ c$\backslash \}\backslash )\backslash (\backslash$mathrm$\{$L$\}=\backslash$left$\backslash \{\backslash$mathrm$\{$wc$\}^\{\backslash$mathrm$\{$k$\}\}\backslash$mid $\backslash$mathrm$\{$w$\}\backslash$in$\backslash \{\backslash$mathrm$\{$a$\},\backslash$mathrm$\{$b$\}\backslash \}^\{*\}\backslash$right$.\backslash )$ and $\backslash (\backslash$left$.\backslash$mathrm$\{$k$\}=|\backslash$mathrm$\{$w$\}|\backslash$right$\backslash \}\backslash )$

$($Hint: It is only asking for the $\backslash$# of c$\text{'}$s $=$ total number of a$\text{'}$s $+$ b$\text{'}$s$)$

$($i$)$ Describe your algorithm

$($ii$)$ Give the description as a complete $7-$tuple with a transition diagram

$($iii$)$ Show configuration sequences on babbcccc leading to acceptance.