
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
We define a beehive of order  $n$  as follows:

a beehive of order 1 is one hexagon

To construct a beehive of order  $n$ , take a beehive of order  $n-1$  and draw a layer of hexagons in the exterior of these hexagons. See diagram for examples of  $n=2,3$ Initially some hexagons are infected by a virus. If a hexagon has been infected, it will always be infected. Otherwise, it will be infected if at least 5 out of the 6 neighbours are infected.

Let  $f(n)$  be the minimum number of infected hexagons in the beginning so that after a finite time, all hexagons become infected. Find  $f(n)$ . 

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