
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  $ABC$  has  $AB = 2$ ,  $BC = 3$ ,  $CA = 4$ , and circumcenter  $O$ . If the sum of the areas of triangles  $AOB$ ,  $BOC$ , and  $COA$  is  $\tfrac{a\sqrt{b}}{c}$  for positive integers  $a$ ,  $b$ ,  $c$ , where  $\gcd(a, c) = 1$  and  $b$  is not divisible by the square of any prime, find  $a+b+c$ .

*Proposed by Michael Tang*

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