
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 $C:\frac{x^2}{a^2}+\frac{y^2}{b^2}=1(a>b, 0<b<2)$ with left and right foci $F_{1}$ and $F_{2}$, point $M$ lies on the ellipse, $MF_{2}\perp F_{1}F_{2}$, and the perimeter of $\triangle MF_{1}F_{2}$ is $6$ with an area of $\frac{3}{2}$.<br/>$(1)$ Find the standard equation of the ellipse $C$;<br/>$(2)$ A line $l$ passing through point $F_{2}$ intersects the ellipse at points $A$ and $B$, and intersects the $y$-axis at point $P$. If $\overrightarrow{PA}={λ_1}\overrightarrow{AF_{2}}$ and $\overrightarrow{PB}={λ_2}\overrightarrow{BF_{2}}$, determine whether $\lambda _{1}+\lambda _{2}$ is a constant value. Explain your reasoning.

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