
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
In the rectangular coordinate system, a circle $\Omega: x^{2}+y^{2}+d x+e y+f=0$ (where $d, e, f$ are real numbers) has a diameter $AB$ with $A(20,22)$ and $B(10,30)$. Find the value of $f$.

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