
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
In an  $m \times n$  grid, each square is either filled or not filled. For each square, its *value* is defined as  $0$  if it is filled and is defined as the number of neighbouring filled cells if it is not filled. Here, two squares are neighbouring if they share a common vertex or side. Let  $f(m,n)$  be the largest total value of squares in the grid. Determine the minimal real constant  $C$  such that   $$ \frac{f(m,n)}{mn} \le C $$  holds for any positive integers  $m,n$ *CSJL*

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