
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 $O: x^{2}+y^{2}=2$ and a line $l: y=kx-2$.

    (1) If line $l$ intersects circle $O$ at two distinct points $A$ and $B$, and when $\angle AOB=\frac{\pi}{2}$, find the value of $k$;

(2) If $k=\frac{1}{2}$, let $P$ be a moving point on line $l$, and construct two tangents $PC$ and $PD$ to circle $O$ from $P$, with $C$ and $D$ being the points of tangency. Investigate whether line $CD$ always passes through a fixed point? If it does, find the fixed point, and if not, provide an explanation;

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