
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
The left and right foci of the ellipse $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ are $F_{1}(-1,0)$ and $F_{2}(1,0)$, respectively. A line $l$ passing through $F_{2}$ intersects the ellipse at points $A$ and $B$. When the line $AB$ is perpendicular to the x-axis, $\angle AF_{1}B=\frac{π}{3}$.  
$(1)$ Find the equation of the ellipse.  
$(2)$ If $|F_{2}A|=\lambda |F_{2}B|$, and $1\leqslant \lambda \leqslant 2$, and the midpoint of the line segment $AB$ is $M$, find the range of the length of the line segment $F_{1}M$.

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