
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
Two regular triangular pyramids with the same base are inscribed in the same sphere. Knowing that the side length of the base of the triangular pyramids is \(a\) and the radius of the sphere is \(R\), and that the angles between the lateral faces and the base of the two pyramids are \(\alpha\) and \(\beta\) respectively, find the value of \(\tan (\alpha+\beta)\).

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