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

$(1)$ 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 circle $O$ and line $l$.

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