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

(1) Write the cartesian coordinates of point $P$ and the cartesian equation of curve $C$;

(2) If $Q$ is a moving point on curve $C$, find the minimum distance of 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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