
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 circles. The first circle is inscribed in triangle \(ABC\), and the second circle is tangent to side \(AC\) and the extensions of sides \(AB\) and \(BC\). It is known that these circles are tangent to each other, the product of their radii is 20, and the angle \(BAC\) is \(\arccos \frac{2}{3}\). Find the perimeter of triangle \(ABC\).

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