
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 a line $l: x+y=3$ and a circle $C: (x-a)^{2}+(y-5)^{2}=10$ intersect at points $A$ and $B$, the tangents to circle $C$ at points $A$ and $B$, denoted as $l_{1}$ and $l_{2}$, intersect at point $P(-\frac{1}{2}, \frac{5}{2})$. Find the area of quadrilateral $ACBP$.

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