
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 $f\left( x \right)=a\ln x+\frac{a}{2}x^{2}-\left( a^{2}+1 \right)x$.

(1) Discuss the monotonicity of the function $f\left( x \right)$;

(2) When $a > 1$, let the minimum value of the function $f\left( x \right)$ be $g\left( a \right)$. If $g\left( a \right) < b-\frac{1}{4}\left( 2a^{3}-2a^{2}+5a \right)$ always holds, find the smallest integer $b$ that satisfies the condition.

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