
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 $xOy$, establish a polar coordinate system with the coordinate origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. The parameteric equations of line $l$ are given by $$ \left\{\begin{array}{l}x=1− \frac{2 \sqrt{5}}{5}t \\ y=1+ \frac{ \sqrt{5}}{5}t\end{array}\right.$$ (where $t$ is the parameter); the polar coordinate equation of curve $C_1$ is $\rho=4\cos\theta$; point $P$ lies on curve $C_1$ and its polar angle is $\frac{\pi}{4}$.

1. Find the rectangular coordinate equation of curve $C_1$ and the normal equation of line $l$.
2. If the parameteric equations of curve $C_2$ are given by $$ \left\{\begin{array}{l}x=\cos\alpha \\ y=\sin\alpha\end{array}\right.$$ (where $\alpha$ is the parameter), and curve $C_3$ is obtained by transforming curve $C_2$ with transformation $\tau:\left\{\begin{array}{l}{x}^{{'}}=2x \\ {y}^{{'}}=y\end{array}\right.$, find the maximum distance from the midpoint $M$ of segment $PQ$ to line $l$, where $Q$ is a moving point on curve $C_3$.

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