
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 \( H \) be a regular hexagon of side length \( x \). Call a hexagon in the same plane a "distortion" of \( H \) if and only if it can be obtained from \( H \) by translating each vertex of \( H \) by a distance strictly less than 1. Determine the smallest value of \( x \) for which every distortion of \( H \) is necessarily convex.

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