
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 $f(x) = -x^2 + m\ln(x) \ (m \in \mathbb{R})$.

(1) Find the intervals of monotonicity for the function $f(x)$;

(2) If $m = 2$, the function $f(x)$ and $g(x) = x - \frac{a}{x} \ (a \in \mathbb{R})$ have the same critical points.

   (i) Find the value of the real number $a$;

   (ii) If $x_1, x_2 \in \left[ \frac{1}{e}, 5 \right]$ (where $e$ is the base of the natural logarithm), and the inequality $\frac{f(x_1) - g(x_2)}{t+1} \leqslant 1$ always holds, find the range of the real number $t$.

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