
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 $f\left( x \right)=x\ln x-ax\left( a\in R \right)$.

(1) Find the monotonic intervals of the function $f\left( x \right)$;

(2) Investigate whether there exists a real number such that $f\left( x \right)+a\geqslant 0$ always holds true. If it exists, find the value; if not, explain the reason.

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