
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 line $C_{1}$: $\begin{cases} & x=1+\frac{\sqrt{2}}{2}t \\ & y=\frac{\sqrt{2}}{2}t \end{cases}$ (where $t$ is a parameter), and the curve $C_{2}$: $\begin{cases} & x=r\cos \theta \\ & y=r\sin \theta \end{cases}$ (where $r > 0$, and $\theta$ is a parameter).

$(1)$ When $r=1$, find the coordinates of the intersection points between $C_{1}$ and $C_{2}$;

$(2)$ Let point $P$ be a moving point on the curve $C_{2}$. When $r=\sqrt{2}$, find the coordinates of point $P$ when the distance from $P$ to the line $C_{1}$ is maximized.

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