
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 $xOy$, the curve $C_{1}$ passes through the point $P(a,1)$, with its parametric equations given by 
$$
\begin{cases} 
x=a- \frac { \sqrt {2}}{2}t \\
y=1+ \frac { \sqrt {2}}{2}t
\end{cases}
(t \text{ is the parameter, } a\in\mathbb{R})
$$
Taking the origin as the pole and the positive $x$-axis as the polar axis, the polar equation of the curve $C_{2}$ is $\rho\cos ^{2}\theta+2\cos \theta-\rho=0$.

$(1)$ Write the general equation of curve $C_{1}$ and the Cartesian coordinate equation of curve $C_{2}$;

$(2)$ Given that curve $C_{1}$ and curve $C_{2}$ intersect at points $A$ and $B$ (with $P$ between $A$ 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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