
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
Given the plane rectangular coordinate system $(xOy)$, with the coordinate origin $O$ as the pole and the positive semi-axis of $x$ as the polar axis, a polar coordinate system is established. The polar coordinate equation of the curve $(C_{1})$ is $P=4\cos \theta$, and the parameter equation of the line $(l)$ is $\begin{cases}x=1-\frac{2\sqrt{5}}{5}t \\ y=1+\frac{\sqrt{5}}{5}t\end{cases} (t \text{ is the parameter})$.

1. Find the rectangular coordinate equation of the curve $(C_{1})$ and the general equation of the line $(l)$.
2. If the parameter equation of the curve $(C_{2})$ is $\begin{cases}x=2\cos \alpha \\ y=\sin \alpha\end{cases} (\alpha \text{ is the parameter})$, the polar angle of the point $P$ on the curve $(C_{1})$ is $\frac{\pi}{4}$, and $Q$ is a moving point on the curve $(C_{2})$, find the maximum distance from the midpoint $M$ of $PQ$ to the line $(l)$.

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