
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 $M$: $(x^{2})+{(y-2)}^{2}=1$, with the center at point $M$, suppose points $B,C$ are on the line $l$: $x-2y=0$, and their $x$-coordinates are $t,t+4(t \in \mathbb{R})$, respectively. Point $P$ is on segment $BC$, and a tangent line $PA$ to circle $M$ passes through point $P$, with the tangent point being $A$.

$(1)$ If $t=0$, and $MP=\sqrt{5}$, find the equation of line $PA$;

$(2)$ The circle passing through points $A,P,M$ has its center at $D$, find the minimum length of segment $DO$, denoted as $L(t)$.

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