
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 a line $l$ in the Cartesian coordinate system $xOy$ with the parametric equation $\begin{cases} x=a+t\cos \theta \\ y=t\sin \theta \end{cases}$ (where $t$ is the parameter, $\theta$ is the angle of inclination), and taking the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis to establish a polar coordinate system, the equation of the curve in the polar coordinate system is $\rho-\rho\cos^2\theta-4\cos\theta=0$.  
$(1)$ Write the Cartesian coordinate equation of curve $C$;  
$(2)$ For point $Q(a,0)$, if line $l$ intersects curve $C$ at points $A$ and $B$, find the value that makes $\dfrac{1}{|QA|^2}+ \dfrac{1}{|QB|^2}$ a constant value.

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