
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 rectangular coordinate system $xOy$, the parametric equations of the curve $C_1$ are given by $\begin{cases} x=2\cos\phi \\ y=\sin\phi \end{cases}$ ($\phi$ is a parameter). Establish a polar coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis. The curve $C_2$ is a circle with a polar coordinate of the center $(\sqrt{7}, \frac{\pi}{2})$ and passes through the pole.

1. Find the polar equation of the curve $C_1$ and the ordinary equation of the curve $C_2$.
2. Given that the ray $\theta=\frac{\pi}{6}$ ($\rho \geq 0$) intersects the curves $C_1$ and $C_2$ at points $A$ and $B$ (point $B$ is different from the coordinate origin $O$), find the length of the line segment $AB$.

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