
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
Let point $F$ be the focus of the parabola $C: x^{2}=2py\left(p \gt 0\right)$. The line passing through point $F$ with a slope of $\sqrt{5}$ intersects $C$ at points $A$ and $B$ with $S_{\triangle AOB}=2\sqrt{6}$ (where $O$ is the origin).<br/>$(1)$ Find the equation of the parabola $C$;<br/>$(2)$ Through point $E(0,2)$, draw two lines $l_{1}$ and $l_{2}$ with slopes $k_{1}$ and $k_{2}$ respectively, intersecting the parabola $C$ at points $P$, $Q$ and $R$, $S$. Given that $|EP|\cdot |EQ|=|ER|\cdot |ES|$, is there a real number $\lambda$ such that $k_{1}+\lambda k_{2}$ is a constant value? If yes, find the value of $\lambda$; if not, explain why.

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