
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
Triangle  $ABC$  satisfies  $\tan A \cdot \tan B = 3$  and  $AB = 5$ . Let  $G$  and  $O$  be the centroid and circumcenter of  $ABC$  respectively. The maximum possible area of triangle  $CGO$  can be written as  $\frac{a\sqrt{b}}{c}$  for positive integers  $a$ ,  $b$ , and  $c$  with  $a$  and  $c$  relatively prime and  $b$  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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