
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
Consider the function $f(x)=\ln{x}-ax+\frac{1-a}{x}-1$ where $0<a<1$.
1. Determine the intervals of monotonicity for the function $f(x)$.
2. When $a=\frac{1}{3}$, let the function $g(x)=x^2-2bx-\frac{5}{9}$, suppose for every $x_1 \in [1,2]$, there exists $x_2 \in [0,1]$ such that $f(x_1) \geq g(x_2)$ holds. Find the range of the real number $b$.

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