
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
Let $f(x)=(a-a^{2})x+\ln x-\frac{1}{x}$, where $a\in \mathbb{R}$.  
$(1)$ Discuss the monotonicity of the function $f(x)$.  
$(2)$ When $a=1$, let $g(x)=xf(x)+x^{2}+1$. Does there exist an integer $t$ such that the inequality $t\geqslant g(x)$ has a solution for $x$? If so, find the smallest value of $t$; if not, explain why.

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