Given the following context$-$free grammar in Chomsky Normal Form $($CNF$)$ and the sentence: $$the cat chased the mouse with a stick.$$

$1.$ Construct the CYK parsing table for the sentence using the given grammar.

$2.$ Show one possible parse tree for the sentence based on the CYK table.

$```$

S $\}->\backslash$mathrm$\{$ NP VP

VP $->$ V NP $|$ V PP

PP $->$ P NP

NP $->$ Det N $|$ N

V $->$ "chased" $|$ "saw"

P $+$ "with"

D $+"$a$"|$ "the"

N $->$ "cat" $|$ "mouse" $|$ "stick"

$```$