
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 $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ passing through the point $Q(1, \frac{3}{2})$ with eccentricity $\frac{1}{2}$, and passing through the point $P(1,2)$, two lines $l_{1}$ and $l_{2}$ are drawn with a sum of slopes equal to $0$. $l_{1}$ intersects $C$ at points $A$ and $B$, and $l_{2}$ intersects $C$ at points $M$ and $N$. <br/> $(1)$ Find the equation of the ellipse $C$; <br/> $(2)$ Determine if there exists a real number $\lambda$ such that $|PA| \cdot |PB| = \lambda |PM| \cdot |PN|$. If it exists, find the value of $\lambda$, if not, explain the reason.

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