
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 $xOy$, there are two circles $C_1$: $(x-1)^2+y^2=1$ and $C_2$: $(x-3)^2+y^2=9$. Using the origin as the pole and the positive half-axis of $x$ as the polar axis, establish the polar coordinate system.
(1) Find the polar equations for $C_1$ and $C_2$.
(2) Consider the curve $C_3$, which is described by the parametric equations $\begin{cases} x=t\cos \alpha \\ y=t\sin \alpha \end{cases}$ (with $t$ as the parameter and $t\neq 0$). If $C_3$ intersects with circles $C_1$ and $C_2$ at points $A$ and $B$ respectively, find the maximum value of the area $S_{\triangle ABC_2}$.

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