
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 the curve $C_1$: $x + \sqrt{3}y = \sqrt{3}$ and $C_2$: $\begin{cases} x= \sqrt{6}\cos \varphi \\ y= \sqrt{2}\sin \varphi \end{cases}$ (with $\varphi$ as the parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, and the same unit of length is taken in both coordinate systems.

$(1)$ Convert the equations of curves $C_1$ and $C_2$ into polar coordinate equations;

$(2)$ Suppose $C_1$ intersects the $x$, $y$ axes at points $M$, $N$ respectively, and the midpoint of segment $MN$ is $P$. If the ray $OP$ intersects $C_1$, $C_2$ at points $P$, $Q$ respectively, find the distance between points $P$ and $Q$.

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