
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 two points $A(\sqrt{3}, 0)$ and $C(-\sqrt{3}, 0)$, a moving point $Q$ always satisfies $|AQ| + |CQ| = 4$. $O$ represents the coordinate origin.

1. Find the equation of the trajectory $E$ of point $Q$ when point $P$ moves on the circle.
2. Let the line passing through point $B(0, -2)$ intersect $E$ at points $M$ and $N$. Find the equation of the line when the area of $\triangle OMN$ is $1$.

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