
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
Three concentric circles have radii  $3$ ,  $4$ , and  $5$ .  An equilateral triangle with one vertex on each circle has side length  $s$ .  The largest possible area of the triangle can be written as  $a+\frac{b}{c}\sqrt{d}$ , where  $a,b,c$  and  $d$  are positive integers,  $b$  and  $c$  are relatively prime, and  $d$  is not divisible by the square of any prime.  Find  $a+b+c+d$ .

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