
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 rectangular coordinate system $xoy$, curve $C_{1}$ passes through point $P(a, 1)$, and its parametric equations are $\begin{cases} x = a + \sqrt{2}t \\ y = 1 + \sqrt{2}t \end{cases}$ ($t$ is the parameter, $a \in \mathbb{R}$). Establish a polar coordinate system with $O$ as the pole and the non-negative half of the $x$-axis as the polar axis. The polar equation of the curve $C_{2}$ is $\rho \cos^2 \theta + 4\cos \theta - \rho = 0$.

(I) Find the Cartesian equation of curve $C_{1}$ and the polar equation of curve $C_{2}$;

(II) Given that 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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