
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 $f(x) = e^{x} - 3x^{2} + ax$, the equation of the tangent line to its graph at $x=1$ is $y = (e-2)x + b$.  
$(1)$ Find the values of $a$ and $b$;  
$(2)$ If the inequality $f(x) > m$ holds for all $x \in [1, +\infty)$, find the maximum integer value of $m$. (Reference data: $\ln 10 \approx 2.3$)

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