
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 a function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(f(x)-x^2+x) = f(x)-x^2+x$,
1. If $f(2) = 3$, find $f(1)$; also, if $f(0) = a$, find $f(a)$;
2. Assume there exists exactly one real number $x$ such that $f(x) = x$, find the analytic expression of function $f(x)$.

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