
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
Let $n$ be the number of points $P$ inside a circle with radius $2$, such that the sum of squares of the distances from $P$ to the endpoints of a given diameter is $8$. Additionally, the distance from $P$ to the center of the circle is $1$. Determine the value of $n$.

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