
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 $E: \frac{x^{2}}{4}+y^{2}=1$, let $P$ be any point on the ellipse. Draw a vertical line $PD$ through $P$, with $D$ as the foot of the perpendicular. Let $M$ be a point such that $\overrightarrow{DM}=2\overrightarrow{DP}$. Find the locus of point $M$, denoted as curve $C$.
Additionally, let a line pass through point $B_{1}(0,1)$, intersecting ellipse $E$ at points $A_{1}$ and $B_{1}$, and intersecting curve $C$ at points $A_{2}$ and $B_{2}$. Determine $|A_{2}B_{2}|$ when $|A_{1}B_{1}|$ is at its maximum.

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