
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 functions $f(x) = x\ln{x}$ and $g(x) = -x^2 + ax - 2$,

(1) Find the minimum value of the function $f(x)$ on the interval $[t, t+1]$ where $t > 0$.

(2) There exists $x_0 \in [1, e]$ such that $f(x_0) \ge g(x_0)$ is satisfied. Find the range of real numbers $a$.

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