
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 $xOy$, the parametric equation of curve $C$ is $\begin{cases} x=\cos \alpha \\ y=\sin \alpha \end{cases}$ ($\alpha$ is the parameter). By keeping the $x$-coordinate of each point on $C$ unchanged and doubling the $y$-coordinate, we obtain curve $C_1$. Establish a polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis.  
$(1)$ Find the polar equation of $C_1$;  
$(2)$ Let $M$, $N$ be two points on $C_1$, if $OM \perp ON$, find the value of $\dfrac {1}{|OM|^{2}}+ \dfrac {1}{|ON|^{2}}$.

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