
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
Consider an ellipse $\dfrac{x^2}{m} + y^2 = 1$ ($m > 1$) and a hyperbola $\dfrac{x^2}{n} - y^2 = 1$ ($n > 0$) that share the same foci $F_1$ and $F_2$. Let $P$ be a point of intersection between the two curves. Calculate the area of the triangle $\triangle F_1PF_2$.

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