
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
Given two sides $a$ and $b$ of a triangle, how can the third side $c$ be chosen such that the points of tangency of the inscribed and escribed circles with this side divide it into three equal segments? For which values of $a$ and $b$ does such a side $c$ exist?

(Consider the exscribed circle that is tangent to side $c$ and the extensions of sides $a$ and $b$.)

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