
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 parameterized equation of line $l_1$ is given by $\begin{cases}x=t-\sqrt{3} \\ y=kt\end{cases} (t\text{ is a parameter})$, and the parameterized equation of line $l_2$ is given by $\begin{cases}x=\sqrt{3}-m \\ y=\frac{m}{3k}\end{cases} (m\text{ is a parameter})$. Let point $P$ be the intersection of lines $l_1$ and $l_2$. As $k$ varies, the locus of point $P$ forms a curve $C_1$.

(1) Find the Cartesian equation of curve $C_1$;
(2) Establish a polar coordinate system with the coordinate origin as the pole and the positive half of the $x$-axis as the polar axis. The polar equation of line $C_2$ is given by $p\sin\left(\theta+\frac{\pi}{4}\right)=4\sqrt{2}$. Let point $Q$ be a moving point on curve $C_1$. Find the minimum distance between point $Q$ and line $C_2$.

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