
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 $E$: $x^{2}=2py(p > 0)$, and the line $y=kx+2$ intersects $E$ at points $A$ and $B$, and $\overrightarrow{OA} \cdot \overrightarrow{OB}=2$, where $O$ is the origin.  
$(1)$ Find the equation of the parabola $E$;  
$(2)$ When $k=1$, find the length of the chord $|AB|$

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