
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 $xOy$, with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of line $l_1$ is $\rho =\dfrac{-1}{{\sin }\theta +a{\cos }\theta }$, and the polar equation of line $l_2$ is $\rho =\dfrac{1}{{\cos }\theta -a{\sin }\theta }$.

$(1)$ Find the equation of the trajectory $C$ of the intersection point of lines $l_1$ and $l_2$;

$(2)$ If there exist $4$ points on curve $C$ that are equidistant from line $l_1$, find the range of values for the real number $a$.

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