
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 $$f(x) = x \ln x$$ and $$g(x) = \int_{0}^{x} (3t^{2} + 2at - 1) \, dt$$,  
(1) Determine the intervals on which the function $f(x)$ is monotonic;  
(2) Find the minimum value of the function $f(x)$ on the interval $[t, t+2]$ where $t > 0$;  
(3) For any $x \in (0, +\infty)$, if $2f(x) \leq g'(x) + 2$ always holds, determine the range of the real number $a$.

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