
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 two tangents $l_1$ and $l_2$ from a point $P$ on the line $x+y=0$ to the circle $(x+1)^2+(y-5)^2=2$, let $A$ and $B$ be the points of tangency. When the lines $l_1$ and $l_2$ are symmetric about the line $y=-x$, calculate the angle $\angle APB$.

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