
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
Let $F\left( \frac{1}{2},0\right)$, point $A$ is on the $x$-axis, and point $B$ is on the $y$-axis, and $\overrightarrow{AM}=2\overrightarrow{AB}$, $\overrightarrow{BA}\cdot \overrightarrow{BF}=0$.
$(1)$ When point $B$ moves on the $y$-axis, find the equation of the trajectory $E$ of point $M$;
$(2)$ Suppose point $P$ is a moving point on trajectory $E$, points $R$, $N$ are on the $y$-axis, and the circle $(x-1)^{2}+y^{2}=1$ is tangent to $\triangle PRN$, find the minimum value of the area of $\triangle PRN$.

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