
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 the Cartesian coordinate system $(xOy)$, the origin coincides with the pole of the polar coordinate system $(OX)$, and the positive semi-axis of the $x$-axis coincides with the polar axis. The unit lengths are the same. The parametric equations of the curve $C$ in the Cartesian coordinate system are given by $x=2m\cos \phi$, $y=n\sin \phi$, where $m,n$ are constants and $\phi$ is the parameter.

1. When $m=n=1$, find the area of the triangle $\triangle AOB$ formed by the points $A$ and $B$ where the curve $C$ intersects the rays $\theta=\frac{\pi}{4}$ and $\theta=-\frac{\pi}{4}$ in the polar coordinate system.
2. When $m=1$ and $n=2$, the parametric equations of the line $l$ in the Cartesian coordinate system are given by $x=t-\sqrt{3}$, $y=\sqrt{3}t+1$, where $t$ is the parameter. Find the coordinates of the intersection point of the curve $C$ and the line $l$.

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