
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 the center of ellipse $C$ is at the origin $O$, and one of its foci $F$ is at $(\sqrt{3},0)$. Points $A_{1}$ and $A_{2}$ are the left and right vertices of the ellipse, and point $B$ is the top vertex of the ellipse. The area of $\triangle OFB$ is $\frac{{\sqrt{3}}}{2}$. Point $M$ is a moving point on the ellipse $C$ in the first quadrant.
1. Find the standard equation of ellipse $C$.
2. If the slopes of lines $MA_{1}$ and $MA_{2}$ are denoted as $k_{1}$ and $k_{2}$ respectively, and $k_{1}+k_{2}=-\frac{3}{4}$, find the coordinates of point $M$.
3. Let line $MA_{1}$ intersect the $y$-axis at point $P$, and line $MA_{2}$ intersect the $y$-axis at point $Q$. If $\overrightarrow{PB}=λ\overrightarrow{BQ}$, find the range of real number $\lambda$.

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