
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
Coordinate Systems and Parametric Equations (4-4)

In the rectangular coordinate system $xoy$, there is a point $P(a, 1)$ and a curve $C_1$ with the parametric equations $x = a + \sqrt{2}t$, $y = 1 + \sqrt{2}t$ ($t$ is the parameter, $a \in \mathbb{R}$). Establish a polar coordinate system with $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis. The polar equation of the curve $C_2$ is $\rho \cos^2 \theta + 4 \cos \theta - \rho = 0$.

(1) Find the Cartesian equation of the curve $C_1$ and the polar equation of the curve $C_2$ in rectangular coordinates.

(2) It is known that the curves $C_1$ and $C_2$ intersect at points $A$ and $B$, and $|PA| = 2|PB|$. Find the value of the real number $a$.

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