
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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