
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 $F_{1}$ and $F_{2}$ are the two foci of the ellipse $C$: $\frac{{y}^{2}}{{a}^{2}}+\frac{{x}^{2}}{{b}^{2}}=1(a>b>0)$, where $A$ is the right vertex, $D$ is the midpoint of $AF_{2}$, and $F_{1}D\bot AF_{2}$. The line $F_{1}D$ intersects $C$ at points $M$ and $N$, and the perimeter of $\triangle AMN$ is $28$. Find the length of the minor axis of ellipse $C$.

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