
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
Find the smallest real  $\alpha$ , such that for any convex polygon  $P$  with area  $1$ , there exist a point  $M$  in the plane, such that the area of convex hull of  $P\cup Q$  is at most  $\alpha$ , where  $Q$  denotes the image of  $P$  under central symmetry with respect to  $M$ .

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