
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 that point $P(-1, \frac{3}{2})$ is on the ellipse $E$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $E$, respectively, and $O$ is the origin. $PF_{1}$ is perpendicular to the x-axis.

$(1)$ Find the equation of the ellipse $E$;  
$(2)$ Let $A$, $B$ be two points on the ellipse $E$ that satisfy $\overrightarrow{PA} + \overrightarrow{PB} = \lambda \overrightarrow{PO} (0 < \lambda < 4$, and $\lambda \neq 2)$, find the slope of line $AB$;  
$(3)$ Under the condition of $(2)$, when the area of $\triangle PAB$ reaches its maximum value, find the value of $\lambda$.

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