
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 curve $C_1$ is defined by $\begin{cases}x=1+ \frac{ \sqrt{3}}{2}t; \\ y=1+ \frac{1}{2}t\end{cases}$ (where $t$ is the parameter), and in the polar coordinate system with $O$ as the pole and the positive $x$-axis as the polar axis, the curve $C_2$ is defined by $\rho=2$.

$(1)$ Find the general equations of curve $C_1$ and curve $C_2$;

$(2)$ Suppose curve $C_1$ intersects curve $C_2$ at points $A$ and $B$, calculate the product of the distances from point $P(1,1)$ to points $A$ and $B$.

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