
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 circle $C$: $(x-3)^2+(y-5)^2=5$, line $l$ passes through the center of the circle and intersects circle $C$ at points $A$ and $B$, and intersects the $y$-axis at point $P$. If $2\overrightarrow{PA}=\overrightarrow{PB}$, then the slope of line $l$ is $k$=$$\_\_\_\_\_\_\_\_$.

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