
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 $C: y^{2}=2px\left(p \gt 0\right)$ passes through the point $Q\left(2,2\right)$.  
$(1)$ Find the standard equation and the directrix equation of the parabola $C$.  
$(2)$ A line passing through the point $M\left(2,0\right)$ intersects the parabola $C$ at points $A$ and $B$. $O$ is the origin. Let the slopes of lines $OA$ and $OB$ be $k_{1}$ and $k_{2}$ respectively. Find the value of $k_{1}k_{2}$.

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