
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 $f(x)=\ln x-ax+a$ $(a\in\mathbb{R})$.
- $(1)$ When $a=1$, find the intervals of monotonicity for the function $f(x)$;
- $(2)$ Let $\lfloor a \rfloor$ denote the greatest integer less than or equal to the real number $a$. If the inequality $f(x) \leqslant x$ always holds, find the maximum value of $\lfloor a \rfloor$.

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