
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 quadratic function
\[ f(x) = x^2 + b x + 8 \quad (b \neq 0) \]

which has two distinct real roots \( x_1 \) and \( x_2 \), and \( x_1 + \frac{1}{x_2} \) and \( x_2 + \frac{1}{x_1} \) are the zero points of a quadratic function \( g(x) \) with the quadratic coefficient being 1. If \( g(1) = f(1) \), find all possible values of \( g(1) \).

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