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

$(1)$ Find the intervals of monotonicity for the function $f(x)$;

$(2)$ If $g(x)=\frac{1}{3}x^3-\ln(x+1)+f(x+2)$ satisfies: for any $x_1, x_2 \in [0,1]$, $|g(x_1)-g(x_2)|\leqslant 1$ always holds, determine the range of the real number $k$.

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