
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
The excircle of a triangle  $ABC$  corresponding to  $A$  touches the lines  $BC,CA,AB$  at  $A_1,B_1,C_1$ , respectively. The excircle corresponding to  $B$  touches  $BC,CA,AB$  at  $A_2,B_2,C_2$ , and the excircle corresponding to  $C$  touches  $BC,CA,AB$  at  $A_3,B_3,C_3$ , respectively. Find the maximum possible value of the ratio of the sum of the perimeters of  $\triangle A_1B_1C_1$ ,  $\triangle A_2B_2C_2$  and  $\triangle A_3B_3C_3$  to the circumradius of  $\triangle ABC$ .

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