
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 through the point $M(1, -1)$, two tangents to the parabola $E$ are drawn, touching the parabola at points $A$ and $B$ respectively. The slope of line $AB$ is $\frac{1}{2}$.  
(1) Find the standard equation of the parabola $E$;  
(2) A line $l: y = kx + m$ ($m \in (2, 4]$) is tangent to the circle $x^2 + (y - 1)^2 = 1$ and intersects the parabola at points $P$ and $Q$. If there exists a point $C$ on the parabola such that $\overrightarrow{OC} = \lambda (\overrightarrow{OP} + \overrightarrow{OQ})$ ($\lambda > 0$), find the range of $\lambda$.

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