
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 that the ellipse $\dfrac{{x}^{2}}{3{m}^{2}}+ \dfrac{{y}^{2}}{5{n}^{2}}=1$ and the hyperbola $\dfrac{{x}^{2}}{2{m}^{2}}- \dfrac{{y}^{2}}{3{n}^{2}}=1$ share common foci,

(1) Find the equations of the asymptotes of the hyperbola;

(2) Line $l$ passes through a focus and is perpendicular to the $x$-axis. If the area of the triangle formed by line $l$ and the asymptotes of the hyperbola is $\dfrac{3}{4}$, find the hyperbola's equation.

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