
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 $C: \dfrac{x^2}{a^2} + \dfrac{y^2}{b^2} = 1$ ($a > b > 0$) with eccentricity $\dfrac{\sqrt{2}}{2}$, the shortest chord through the right focus of the ellipse $C$ has a length of $2$.
1. Find the equation of the ellipse $C$;
2. Given that the left vertex of the ellipse $C$ is $A$, and $O$ is the origin, is there a tangent line $l$ on the circle with diameter $AO$ that intersects the ellipse $C$ at two distinct points $M$ and $N$, such that the product of the slopes of lines $OM$ and $ON$ is $\dfrac{7}{16}$? If such a tangent line $l$ exists, find its equation; if not, explain why.

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