
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 $(xOy)$, curve $C\_1$ passes through point $P(a,1)$, with parameter equations $\begin{cases} X=a+\sqrt{2}t \\ Y=1+\sqrt{2}t \end{cases}$ ($t$ is the parameter, $a \in \mathbb{R}$). In the polar coordinate system established with $O$ as the pole and the non-negative semi-axis of $x$ as the polar axis, the polar equation of curve $C\_2$ is given by $\rho \cos^2 \theta + 4 \cos \theta - \rho = 0$.

(1) Find the ordinary equation of curve $C\_1$ and the rectangular coordinate equation of curve $C\_2$;

(2) If curves $C\_1$ and $C\_2$ intersect at points $A$ and $B$, and $|PA| = 2|PB|$, find the value of the real number $\alpha$.

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