
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 parametric equation of curve $C$ is $\begin{cases} & x=2\sqrt{3}\cos \alpha \\ & y=2\sin \alpha \end{cases}$, where $\alpha$ is the parameter, $\alpha \in (0,\pi)$. In the polar coordinate system with the origin $O$ as the pole and the positive direction of the axis as the polar axis, the polar coordinates of point $P$ are $(4\sqrt{2},\frac{\pi}{4})$, and the polar equation of the line is $\rho \sin (\theta -\frac{\pi}{4})+5\sqrt{2}=0$.

(Ⅰ) Find the Cartesian equation of the line and the standard equation of curve $C$;

(Ⅱ) If $Q$ is a moving point on curve $C$, and $M$ is the midpoint of segment $PQ$. Find the maximum distance from point $M$ to the line.

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