
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 polar coordinate equation of line $l$ as $2\rho \sin\left(\theta - \frac{\pi}{4}\right) = \sqrt{2}$, and the polar coordinates of point $A$ as $\left(2\sqrt{2}, \frac{7\pi}{4}\right)$, find the distance from point $A$ to line $l$.

Also, derive the polar coordinate equation of a circle centered at the origin with radius $r$ from its Cartesian coordinate equation $x^{2}+y^{2}=r^{2}$ $(r > 0)$.

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