
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
The equation \( x^{2}-x+1=0 \) has roots \(\alpha\) and \(\beta\). There is a quadratic function \( f(x) \) that satisfies \( f(\alpha) = \beta \), \( f(\beta) = \alpha \), and \( f(1) = 1 \). What is the expression for \( f(x) \)?

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