
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 eccentricity $\frac{1}{2}$ and passing through the point $M(1,\frac{3}{2})$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Let the left and right vertices of the ellipse $C$ be $A_{1}$ and $A_{2}$, and let $P$ and $Q$ be points on the ellipse $C$ other than $A_{1}$ and $A_{2}$. If the line $PQ$ passes through the point $(\frac{1}{2},0)$, is there a real number $\lambda$ such that ${k}_{A_{1}P}=\lambda{k}_{A_{2}Q}$ always holds true? If yes, find the value of the real number $\lambda$; if not, explain the reason.

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