
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
Given positive integers  $k \ge 2$  and  $m$  sufficiently large. Let  $\mathcal{F}_m$  be the infinite family of all the (not necessarily square) binary matrices which contain exactly  $m$  1's. Denote by  $f(m)$  the maximum integer  $L$  such that for every matrix  $A \in \mathcal{F}_m$ , there always exists a binary matrix  $B$  of the same dimension such that (1)  $B$  has at least  $L$  1-entries; (2) every entry of  $B$  is less or equal to the corresponding entry of  $A$ ; (3)  $B$  does not contain any  $k \times k$  all-1 submatrix. Show the equality
\[\lim_{m \to \infty} \frac{\ln f(m)}{\ln m}=\frac{k}{k+1}.\]

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