
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 $f(x)=(x-a)\ln x-x+a-3$ $(a\in R)$, find:

1. When $a=0$, determine the minimum value of $f(x)$.
2. Discuss the monotonicity of $f'(x)$.
3. When $a=2$, if $\lambda \leq f(x)$ always holds, find the maximum integer value of $\lambda$.

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