
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
Let \( S_{0} \) be a sphere with center at fixed point \( P_{0} \) and radius \( r \), and let \( \pi_{0} \) be a fixed plane. The distance from \( P_{0} \) to \( \pi_{0} \) is \( a > r \). Let \( S_{M} \) be a sphere with center at point \( M \) that is externally tangent to \( S_{0} \) and tangent to \( \pi_{0} \). Let \( \Lambda \) be the set of all possible centers \( M \) of spheres \( S_{M} \) that satisfy these conditions. Suppose \( \pi \) is a plane parallel to \( \pi_{0} \) and intersects \( \Lambda \) at some point. Let \( d_{\Sigma} \) be the distance between the planes \( \pi \) and \( \pi_{0} \). Determine the minimum value \( m \) of \( d_{\Sigma} \).

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