
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 polar coordinate system, the polar equation of curve $C$ is given by $\rho=2\cos \theta+2\sin \theta$ $(0\leqslant \theta < 2\pi)$. Point $M$ has coordinates $\left(1, \frac{\pi}{2}\right)$. A Cartesian coordinate system is established with the pole $O$ as the origin and the polar axis as the positive $x$-axis. It is known that line $l$, defined by the parametric equations $\begin{cases}x= \frac{ \sqrt{2}}{2}t \\ y=1+ \frac{ \sqrt{2}}{2}t\end{cases}$ $(t$ is the parameter$)$, intersects curve $C$ at points $A$ and $B$, with $|MA| > |MB|$.

(1) Find the maximum value of $\rho$ for any point $P\left(\rho,\theta\right)$ on curve $C$ and determine the polar coordinates of $P$ in this case.

(2) Determine $\frac{|MA|}{|MB|}$.

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