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

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