
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 the function $f(x)=x+\ln x- \frac {a-a^{2}}{x}$ $(a\in\mathbb{R})$.  
(1) Discuss the monotonicity of the function $f(x)$.  
(2) When $a=1$, let $g(x)=xf(x)$. Determine whether there exists an integer $t$ such that the inequality $t\geqslant g(x)$ has a solution for $x$. If such $t$ exists, find the smallest value of $t$; if not, provide the reasoning.

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