
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 a circle E tangent to the x-axis, with its center on the positive y-axis, and the chord intercepted by the line x-y=0 is of length $2\sqrt{2}$.
(1) Find the standard equation of circle E;
(2) A line passing through the fixed point P(-3, 0) intersects the circle E at two distinct points M and N. On the line segment MN, a point H distinct from M and N is chosen such that $\frac{|\overrightarrow{PM}|}{|\overrightarrow{PN}|} = \frac{|\overrightarrow{MH}|}{|\overrightarrow{NH}|}$. Find the range of the x-coordinate $x_0$ of the point H.

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