
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 Cartesian coordinate system $xOy$, let $A$ and $B$ be two moving points on circle $C: (x+4)^{2}+(y-a)^{2}=16$, and the distance between $A$ and $B$ is $AB = 2\sqrt{11}$. If there exists a unique point $P$ on the line $l: y=2x$ such that $\overrightarrow{PA} + \overrightarrow{PB} = \overrightarrow{OC}$, find the value of the real number $a$.

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