
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 parametric equation of curve $C_1$ is $$\begin{cases} x=2+\cos\alpha \\ y=4+\sin\alpha\end{cases}$$, in the coordinate system with the origin $O$ as the pole and the positive half-axis of $x$ as the polar axis, the equation of curve $C_2$ is $\rho(\cos\theta-m\sin\theta)+1=0$ ($m$ is a constant).  
(1) Find the Cartesian equations of curves $C_1$ and $C_2$;  
(2) Let point $P$ be the point on $C_1$ that is closest to the $x$-axis. When $C_2$ passes through point $P$, find the value of $m$.

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