
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, it is known that circle $O$: $\rho=\cos \theta+\sin \theta$ and line $l$: $\rho\sin \left( \theta- \frac{\pi}{4} \right)= \frac{ \sqrt{2}}{2}(\rho\geqslant 0,0\leqslant \theta < 2\pi)$.

$(1)$ Find the Cartesian coordinate equations of circle $O$ and line $l$;

$(2)$ When $\theta\in(0,\pi)$, find the polar coordinates of the intersection points of line $l$ and circle $O$.

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