
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
(1) It is known that the foci of a hyperbola are on the y-axis, and the ratio of the length of its real axis to the length of its imaginary axis is 2:3. The hyperbola goes through the point P($\sqrt{6}$, 2). Find the equation of the hyperbola.
(2) The foci of another hyperbola are on the x-axis, its eccentricity is $\frac{5}{3}$, and it passes through point M(-3, 2$\sqrt{3}$). Find the equation of this hyperbola.

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