
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
In the Cartesian coordinate system $xOy$, $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C$: $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 (a > b > 0)$, respectively. $B$ is one endpoint of the minor axis, and $E$ is a point on the ellipse $C$ that satisfies $\overrightarrow{OE} = \overrightarrow{OF_{1}} + \frac{\sqrt{2}}{2} \overrightarrow{OB}$, and the perimeter of $\triangle EF_{1}F_{2}$ is $2(\sqrt{2} + 1)$.  
$(1)$ Find the equation of the ellipse $C$;  
$(2)$ Let point $M$ be a point on the line segment $OF_{2}$, and a line $l$ passing through point $F_{2}$ and not perpendicular to the $x$-axis intersects the ellipse $C$ at points $P$ and $Q$. If $\triangle MPQ$ is an isosceles triangle with $M$ as the vertex, find the range of distances from point $M$ to line $l$.

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