
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 the parabola $C: x^{2}=2py\left(p \gt 0\right)$, the line $l: y=kx+2$ intersects $C$ at points $A$ and $B$, and $OA\bot OB(O$ is the origin).<br/>$(1)$ Find the equation of the parabola $C$;<br/>$(2)$ Let $P\left(2,2\right)$, if the slopes of lines $PA$ and $PB$ are complementary, find the value of $k$.

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