
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
Suppose $4-4$: Coordinate Systems and Parametric Equations
In the rectangular coordinate system $xOy$, the parametric equations of the curve $C$ are: $\begin{cases}x=1+ \sqrt{2}\cos \phi \\ y=1- \sqrt{2}\sin \phi \end{cases}$ ($\phi$ is the parameter), and in the polar coordinate system with $O$ as the pole and the non-negative half of the $x$-axis as the polar axis, the polar equation of the line $l$ is: $2\rho\cos \theta+2\rho\sin \theta-1=0$.
(1) Find the ordinary equations of the curve $C$ and the line $l$.
(2) Let $d$ be the distance from a point on the curve $C$ to the line $l$. Find the maximum value of $d$.

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