
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 ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, with the left and right foci being $F_1$ and $F_2$ respectively, and the top vertex $A(0,b)$. If $\triangle AF_1F_2$ is an equilateral triangle and its perimeter is $6$.
$(1)$ Find the standard equation of the ellipse $C$ and its eccentricity;
$(2)$ Let $O$ be the origin, and $P$ be a moving point on the line $F_1A$. Find the minimum value of $|PF_2|+|PO|$, and the coordinates of point $P$ at this time.

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