
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 the hyperbola $C$: $x^{2} - y^{2} = 1$.
$(1)$ Find the equation of the circle with its center at the right focus of the hyperbola $C$ and tangent to the asymptotes of $C$;  
$(2)$ If a line passing through the point $P(0,-1)$ intersects the right branch of the hyperbola $C$ at two distinct points $M$ and $N$, find the range of the $y$-intercept $t$ of the perpendicular bisector $l$ of line segment $MN$.

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