
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
If an ellipse $\frac{x^{2}}{10} + \frac{y^{2}}{m} = 1$ and a hyperbola $x^{2} - \frac{y^{2}}{b} = 1$ have the same foci, and they intersect at point $P\left(\frac{\sqrt{10}}{3}, y\right)$, find the equations of the ellipse and the hyperbola.

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