
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 $f(x) = \ln x - ax + \frac{1-a}{x} + 1$ ($a \in \mathbb{R}$).

(Ⅰ) Find the interval of monotonic increase for the function $f(x)$.

(Ⅱ) When $a \in \left( \frac{1}{3}, 1 \right)$, if for any $t \in [2, 3]$, the minimum value of the function $f(x)$ for $x \in (0, t]$ is $f(t)$, find the range of the real number $a$.

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