
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
A coordinate system and parametric equations (4-4): In the rectangular coordinate system $xOy$, an polar coordinate system is established with $O$ as the pole and the positive semi-axis of $x$ as the polar axis. The parametric equations of curve $M$ are given by $\begin{cases} x=1+\cos\varphi \ y=1+\sin\varphi \end{cases}$ $(\varphi\text{ is the parameter})$. $l_1$ and $l_2$ are two lines passing through point $O$, intersecting curve $M$ at points $A$, $B$ and $C$, $D$ respectively, and the slope angle of $l_1$ is $\alpha$, $\angle AOC=\frac{\pi}{6}$.

(1) Find the polar coordinate equation of $l_1$ and curve $M$.

(2) When $\alpha\in\left(0,\frac{\pi}{6}\right]$, find the maximum value of the sum of the distances from point $O$ to the four points $A$, $B$, $C$, and $D$.

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