
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 the origin as the pole and the non-negative half-axis of $x$ as the polar axis, a polar coordinate system is established. It is known that the parametric equation of line $l$ is $\begin{cases} x= \sqrt {2}+t \\ y=t\end{cases}$ (where $t$ is the parameter), and the polar equation of circle $C$ is $\rho=1$.

$(1)$ Find the number of common points between line $l$ and circle $C$;

$(2)$ In the Cartesian coordinate system, circle $C$ undergoes a scaling transformation $\begin{cases} \overset{x'=x}{y'=2y}\end{cases}$ to obtain curve $C′$. Let $M(x,y)$ be a point on curve $C′$, find the maximum value of $4x^{2}+xy+y^{2}$, and the coordinates of the corresponding point $M$.

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