
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 lateral faces of a triangular pyramid are equal in area and form angles $\alpha, \beta$, and $\gamma$ with the base. Find the ratio of the radius of the sphere inscribed in this pyramid to the radius of the sphere that touches the base of the pyramid and the extensions of the lateral sides.

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