
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 is curve C<sub>1</sub>: 
$$
\begin{cases} 
x=-2+\cos\alpha \\ 
y=-1+\sin\alpha 
\end{cases}
$$
(where $\alpha$ is a parameter). In the polar coordinate system with the origin O as the pole and the positive x-axis as the initial line, there is curve C<sub>2</sub>: $\rho\cos\theta - 3 = 0$. Let point P be a moving point on curve C<sub>1</sub>. 
(1) Find the maximum distance from point P to the curve C<sub>2</sub>;  
(2) If curve C<sub>3</sub>: $\theta = \frac{\pi}{4}$ intersects curve C<sub>1</sub> at points A and B, find the area of triangle $\triangle ABC_1$.

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