
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 $xOy$, with $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, a polar coordinate system is established. The polar coordinates of point $P$ are $(3, \frac{\pi}{4})$. The parametric equation of curve $C$ is $\rho=2\cos (\theta- \frac{\pi}{4})$ (with $\theta$ as the parameter).

(Ⅰ) Write the Cartesian coordinates of point $P$ and the Cartesian coordinate equation of curve $C$;  
(Ⅱ) If $Q$ is a moving point on curve $C$, find the minimum distance from the midpoint $M$ of $PQ$ to the line $l$: $2\rho\cos \theta+4\rho\sin \theta= \sqrt{2}$.

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