
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  $ \mathcal{F}$  be a family of hexagons  $ H$  satisfying the following properties:

i)  $ H$  has parallel opposite sides.

ii) Any 3 vertices of  $ H$  can be covered with a strip of width 1.

Determine the least  $ \ell\in\mathbb{R}$  such that every hexagon belonging to  $ \mathcal{F}$  can be covered with a strip of width  $ \ell$ .

Note: A strip is the area bounded by two parallel lines separated by a distance  $ \ell$ . The lines belong to the strip, too.

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