
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 Cartesian coordinate system \( xOy \), the parametric equation of the line \( l \) is \(\begin{cases} x=1+t\cos \alpha \\ y=2+t\sin \alpha \end{cases}\) (\(t\) as the parameter), in the polar coordinate system (using the same unit length as the Cartesian coordinate system \(xOy\)), with the origin \(O\) as the pole and the non-negative half-axis of the \(x\)-axis as the polar axis, the equation of the circle \(C\) is \( \rho = 6\sin \theta \).
(1) Find the Cartesian coordinate equation of circle \(C\);
(2) If point \(P(1,2)\) is given, and circle \(C\) intersects with line \(l\) at points \(A\) and \(B\), find the minimum value of \(|PA|+|PB|\).

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