
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 $f(x)=ax-b\ln x$, the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$ is $y=x+1$.  
$(1)$ Find the intervals of monotonicity for the function $f(x)$;  
$(2)$ For any $x \geqslant 1$, $f(x) \geqslant kx$ always holds, find the range of values for $k$.

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