
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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