
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 curve $C\_1$ are given by $\begin{cases} x=2\cos \alpha \ y=2+2\sin \alpha \end{cases}$, where $\alpha$ is the parameter. $M$ is a moving point on $C\_1$, and point $P$ satisfies $\overrightarrow{OP} =2\overrightarrow{OM}$. The trajectory of point $P$ forms curve $C\_2$.

1. Find the equation of $C\_2$;
2. In the polar coordinate system with $O$ as the pole and the positive semi-axis of $x$ as the polar axis, the ray $\theta=\frac{\pi}{3}$ intersects $C\_1$ at point $A$ and $C\_2$ at point $B$ (both intersections are distinct from the pole). Find $|AB|$;
3. If line $l$: $\begin{cases} x=4-\sqrt{3}t \ y=-t \end{cases}$ (with $t$ as the parameter) intersects curve $C\_2$ at points $E$ and $F$, and the midpoint of $EF$ is $G$, while point $H(4,0)$, find $|HG|$.

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