$(3)$ Create a PDA that recognizes the following context free language with terminals $\{a,b,c\}$

$L=\{w{c}^k|win\{a,(b){\}}^{**}$ and $k=|w|\}$

$($Hint: It is only asking for the # 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.