
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 curve $f(x) = x^{3} + 2x^{2} + mx + n$ has a tangent line equation $y = 4x + 3$ at $x = 0$.  
$(1)$ Find the values of $m$ and $n$.  
$(2)$ Let $k$ be an integer. For $x > 1$, the inequality $f(x+x\ln x) > f(k(x-1))$ always holds. Find the maximum value of $k$.

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