
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 $M: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$ with eccentricity $\frac{\sqrt{3}}{2}$ and the length of the minor axis is $2$.

 $(1)$ Find the standard equation of the ellipse $M$

 $(2)$ If the line $l$ passing through the point $(0,2)$ intersects the ellipse $M$ at points $P$ and $Q$, satisfying $\overrightarrow{OP} \cdot \overrightarrow{OQ} = 0$, find the equation of $l$

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