
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
Distinct planes $p_1, p_2, ..., p_n$ tangent to the surface of a sphere $S$. On this sphere, defined points $A_1, A_2, ..., A_{12}$, are spaced equidistantly along the equator. Each of the planes intersects another sphere $T$, concentric with $S$. The intersection forms chords on $T$ that join pairs of points among $A_1, ..., A_{12}$. Determine the difference between the maximum and minimum possible values of $n$.

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