
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 real number $m > 1$, fixed points $A(-m, 0)$, $B(m, 0)$, and a moving point $S$, the product of the slopes of the lines connecting point $S$ with points $A$ and $B$ is $-\frac{1}{m^2}$.

$(1)$ Find the equation of the trajectory $C$ of the moving point $S$ and identify the type of curve it represents;

$(2)$ If $m = \sqrt{2}$, what value should $t$ take so that the line $l: 2x - y + t = 0$ ($t > 0$) has exactly one intersection point with curve $C$?

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