
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 the function $f(x)=\lg(ax-3)$, and its graph passes through the point $(2,0)$. If $k$ is a positive integer, then the maximum value of $k$ that makes the inequality $2f(x) > \lg(kx^{2})$ have a solution in the interval $[3,4]$ is ____.

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