
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 an ellipse $C$ centered at the origin, the two foci of ellipse $C$ and ellipse $C_1: 2x^2+3y^2=72$ are the four vertices of a square, and ellipse $C$ passes through point $A(\sqrt{3}, -2)$.
(1) Find the standard equation of ellipse $C$;
(2) Given $P$ is any point on ellipse $C$, and $Q(0, t)$, find the minimum value of $|PQ|$.

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