
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
3. Let  $ABC$  be a triangle with  $AB=30$ ,  $BC=14$ , and  $CA=26$ . Let  $N$  be the center of the equilateral triangle constructed externally on side  $AB$ . Let  $M$  be the center of the square constructed externally on side  $BC$ . Given that the area of quadrilateral  $ACMN$  can be expressed as  $a+b\sqrt{c}$  for positive integers  $a$ ,  $b$  and  $c$  such that  $c$  is not divisible by the square of any prime, compute  $a+b+c$ . 

*Proposed by winnertakeover*

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