
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 \(f(x)=\ln x-ax+\frac{1-a}{x}-1\).  
(i) When \(a=1\), find the equation of the tangent line to the curve \(f(x)\) at \(x=1\).  
(ii) When \(a=\frac{1}{3}\), determine the intervals of monotonicity for the function \(f(x)\).  
(iii) Under the condition in (ii), suppose a function \(g(x)=x^{2}-2bx-\frac{5}{12}\), such that for every \(x_{1}\) in the interval \([1,2]\), there exists an \(x_{2}\) in the interval \([0,1]\) that satisfies \(f(x_{1}) \geq g(x_{2})\), determine the range of values for the real number \(b\).

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