
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 a triangle $ABC$ with vertices $A$ and $B$ at coordinates $(0, \sqrt{3})$ and $(0, -\sqrt{3})$ respectively, and the product of slopes of lines $AC$ and $BC$ is equal to $m (m\neq 0)$.
1. Find the equation of the trajectory $M$ of vertex $C$ and determine what kind of curve the trajectory $M$ is.
2. When $m=-\frac{3}{4}$, a point $P(1,t)$ lies on curve $M$, and $P$ is in the first quadrant. Draw two lines through $P$ that intersect curve $M$ at points $E$ and $F$. If the slopes of lines $PE$ and $PF$ are negatives of each other, is the slope of line $EF$ a constant value? If so, find the constant value. If not, please explain the reason.

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