
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 $f(x) = ax^2 + x \ln x + b$ ($a, b \in \mathbb{R}$), the equation of the tangent at the point $(1, f(1))$ is $3x - y - 4 = 0$. If there exists an integer $k$ such that $f(x) > k$ always holds true, find the maximum value of $k$.

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