
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 regular triangular pyramids \( P-ABC \) and \( Q-ABC \) with the same base \( ABC \) such that both are inscribed in the same sphere. If the angle between the side face and the base of the regular triangular pyramid \( P-ABC \) is \( 45^{\circ} \), find the tangent of the angle between the side face and the base of the regular triangular pyramid \( Q-ABC \).

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