
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 $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b>0)$ with left and right vertices $A_{1}$, $A_{2}$, $A_{2}$, and a focal length of $2\sqrt{5}$, where point $B(0,2)$ lies on the ellipse. <br/>$(Ⅰ)$ Find the equation of $C$;<br/>$(Ⅱ)$ Any line passing through point $P(1,0)$ intersects the ellipse $C$ at points $M$ and $N$ (distinct from $A_{1}$, $A_{2}$, $A_{2}$). The slope of line $A_{1}M$ is $k_{1}$, and the slope of line $A_{2}N$ is $k_{2}$. Is there a constant $\lambda$ such that $k_{1}=\lambda k_{2}$? If so, find the value of $\lambda$; if not, explain why.

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