
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Point  $P$  is situated inside regular hexagon  $ABCDEF$  such that the feet from  $P$  to  $AB$ ,  $BC$ ,  $CD$ ,  $DE$ ,  $EF$ , and  $FA$  respectively are  $G$ ,  $H$ ,  $I$ ,  $J$ ,  $K$ , and  $L$ . Given that  $PG = \frac92$  ,  $PI = 6$ , and  $PK =\frac{15}{2}$  , the area of hexagon  $GHIJKL$  can be written as  $\frac{a\sqrt{b}}{c}$  for positive integers  $a$ ,  $b$ , and  $c$  where  $a$  and  $c$  are relatively prime and  $b$  is not divisible by the square of any prime. Find  $a + b + c$ .

Present the answer in LaTex format: \boxed{Your answer}
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