
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 circle $C$ with the equation $(x-a)^2 + (y-2)^2 = 4$ ($a > 0$) and a line $l$ with the equation $x - y + 3 = 0$, which intersects the circle at points $A$ and $B$:
1. When the chord AB cut by line $l$ on circle $C$ has a length of $2\sqrt{2}$, find the value of $a$.
2. If there exists a point $P$ on the circle such that $\overrightarrow{CA} + \overrightarrow{CB} = \overrightarrow{CP}$, find the value of $a$.

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