
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 $P(a, 1)$ with parametric equations $$\begin{cases} x=a+ \frac { \sqrt {2}}{2}t \\ y=1+ \frac { \sqrt {2}}{2}t\end{cases}$$ where $t$ is a parameter and $a \in \mathbb{R}$. In the polar coordinate system with the pole at $O$ and the non-negative half of the $x$-axis as the polar axis, the polar equation of curve $C_2$ is $\rho\cos^2\theta + 4\cos\theta - \rho = 0$.
(I) Find the Cartesian equation of the curve $C_1$ and the polar equation of the curve $C_2$;
(II) Given that the curves $C_1$ and $C_2$ intersect at points $A$ and $B$, with $|PA| = 2|PB|$, find the value of the real number $a$.

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