
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 circle $M: (x-2)^{2}+y^{2}=4$, point $P(-1,t)$ where $t\in \mathbb{R}$.<br/>$(1)$ If $t=1$, a circle $N$ with radius $1$ passes through point $P$ and is externally tangent to circle $M$, find the equation of circle $N$;<br/>$(2)$ If the two chords passing through point $P$ and intercepted by circle $M$ have lengths of $2\sqrt{3}$ each, intersect the $y$-axis at points $S$ and $T$, $|ST|=\frac{3}{4}$, find $t$.

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