
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
In a triangle $\triangle PMN$ with an area of 1, $\tan M = \frac{1}{2}$ and $\tan N = -2$. Establish an appropriate coordinate system and find the equation of the ellipse with foci at $M$ and $N$ that passes through point $P$.

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