
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 function $f(x)$ satisfies $f(x+1)=x^{2}- \frac{1}{3} f(3)$.

$(1)$ Find the expression for $f(x)$;

$(2)$ Determine whether there exists $q > 0$ such that $g(x)=1-qf(x)-x$ has its range on $[-1,2]$ as $[-4, \frac{17}{8}]$. If it exists, find $q$; if not, explain why.

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