
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 that the function $f(x)=x^{2}+ax+b$ ($a < 0, b > 0$) has two distinct zeros $x_{1}$ and $x_{2}$, and after arranging $-2$, $x_{1}$, and $x_{2}$ appropriately, they can form either an arithmetic progression or a geometric progression, determine the analytical expression of the function $f(x)$.

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