
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, the polar equation of curve $C_1$ is: $3\rho^{2}=12\rho\cos \theta-10$ ($\rho > 0$).
$(1)$ Find the Cartesian equation of curve $C_1$.
$(2)$ The equation of curve $C_2$ is $\frac {x^{2}}{16}+ \frac {y^{2}}{4}=1$. Let $P$ and $Q$ be any points on curves $C_1$ and $C_2$ respectively. Find the minimum value of $|PQ|$.

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