
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
Let  $f$  be a quadratic function which satisfies the following condition. Find the value of  $\frac{f(8)-f(2)}{f(2)-f(1)}$ .

For two distinct real numbers  $a,b$ , if  $f(a)=f(b)$ , then  $f(a^2-6b-1)=f(b^2+8)$ .

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