
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 the ellipse $C$: $\dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1 (a > b > 0)$, with $F(\sqrt{2}, 0)$ as its right focus, and the chord obtained by intersecting the ellipse with the line perpendicular to the $x$-axis passing through $F$ has a length of $2$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ Suppose the line $l$: $y = kx + m (|k| \leqslant \dfrac{\sqrt{2}}{2})$ intersects the ellipse $C$ at points $A$ and $B$. A parallelogram $OAPB$ is formed with $OA$ and $OB$ as adjacent sides, where the vertex $P$ is on the ellipse $C$ and $O$ is the origin. Find the range of values for $|OP|$.

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