
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
Convex equiangular hexagon  $ABCDEF$  has  $AB=CD=EF=1$  and  $BC = DE = FA = 4$ . Congruent and pairwise externally tangent circles  $\gamma_1$ ,  $\gamma_2$ , and  $\gamma_3$  are drawn such that  $\gamma_1$  is tangent to side  $\overline{AB}$  and side  $\overline{BC}$ ,  $\gamma_2$  is tangent to side  $\overline{CD}$  and side  $\overline{DE}$ , and  $\gamma_3$  is tangent to side  $\overline{EF}$  and side  $\overline{FA}$ . Then the area of  $\gamma_1$  can be expressed as  $\frac{m\pi}{n}$  for relatively prime positive integers  $m$  and  $n$ . Compute  $100m+n$ .

*Proposed by Sean Li*

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