
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 an ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1(a>b>0)$ with the right vertex $A$ and the top vertex $B$, and $O$ as the origin. A point $M$ inside the ellipse satisfies $\overrightarrow{OM}=\overrightarrow{MA}$, and $\frac{|BM|}{|AB|}=\frac{\sqrt{6}}{4}$. <br/>$(Ⅰ)$ Find the eccentricity of the ellipse; <br/>$(Ⅱ)$ Point $P$ on the ellipse is in the first quadrant and satisfies $∠AMP=\frac{π}{6}$. Line $PO$ intersects the ellipse at point $Q$, and line $AQ$ intersects the extension of $PM$ at point $D$. If the area of $\triangle PDQ$ is $\frac{5\sqrt{3}}{12}$, find the standard equation of the ellipse.

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