
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 $f(x) = \ln x - ax + \frac{1-a}{x} - 1$,
(1) When $a=1$, find the equation of the tangent line to the curve $f(x)$ at $x=1$;
(2) When $a=\frac{1}{3}$, determine the intervals on which the function $f(x)$ is monotonic;
(3) Under the condition of (2), let $g(x) = x^2 - 2bx - \frac{5}{12}$. If for every $x_1 \in [1,2]$, there exists $x_2 \in [0,1]$ such that $f(x_1) \ge g(x_2)$ holds, find the range of the real number $b$.

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