
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
137.2 ** Quadrilateral $PQRS$ is a rhombus inscribed in the ellipse $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$.
(1) Through origin $O$, draw a perpendicular to side $PQ$ with foot $H$. Find the locus of $H$.
(2) Find the maximum value of $\frac{1}{|OP|}+\frac{1}{|OQ|}$.
(3) Find the maximum and minimum values of the area of $\triangle POQ$.
(4) Find the maximum and minimum values of $|PQ|$.

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