
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 plane Cartesian coordinate system $xOy$, the equation of the line $l$ is $x+y=5$. The circle $M$ has its center at $(3,0)$ and is tangent to the line $l$. Establish a polar coordinate system with the origin as the pole and the positive x-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equation of the circle $M$;<br/>$(2)$ If the ray $θ=α({0＜α＜\frac{π}{2},ρ＞0})$ intersects the circle $M$ at points $A$ and $B$, and $\frac{1}{{|{OA}|}}+\frac{1}{{|{OB}|}}=\frac{1}{7}$, find the Cartesian equation of the line $AB$.

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