
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 $P(a,1)$, and its parametric equations are 
$$
\begin{cases}
x=a+ \sqrt{2}t \\
y=1+ \sqrt{2}t
\end{cases} 
$$
where $t$ is the parameter, $a\in \mathbb{R}$. With $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $p\cos^2\theta+4\cos\theta-p=0$.

$(1)$ Find the general equation of curve $C_1$ and the Cartesian equation of curve $C_2$;

$(2)$ Given that curve $C_1$ intersects curve $C_2$ 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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