
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
For the function $f(x)$, if there exists $x \in \mathbb{R}$ such that $f(x) = x$ holds, then $x$ is called a fixed point of $f(x)$. Given $f(x) = ax^2 + (b+1)x + (b-1)$ ($a \neq 0$), when $a=1$ and $b=-2$, find the fixed points of $f(x)$.

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