
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
C is a non-empty, closed, bounded, convex subset of the plane. Given a support line L of C and a real number 0 ≤ α ≤ 1, let Bᵅ be the band parallel to L, situated midway between L and the parallel support line on the other side of C, and of width α times the distance between the two support lines. What is the smallest α such that ∩ Bᵅ contains a point of C, where the intersection is taken over all possible directions for the support line L?

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