
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 parametric equations: 
$$
\begin{cases} 
x=a+ \sqrt {2}t \\
y=1+ \sqrt {2}t
\end{cases} \quad (t \text{ as the parameter, } a\in \mathbb{R}).
$$
Taking the origin $O$ as the pole and the non-negative half of the $x$-axis as the polar axis, the curve $C_2$ is given by the polar equation:
$$
\rho\cos^{2}\theta+4\cos \theta-\rho=0.
$$
(Ⅰ) Find the standard equation of curve $C_1$ and the Cartesian equation of curve $C_2$;
(Ⅱ) It is known that curves $C_1$ and $C_2$ intersect at points $A$ and $B$, and $|PA|=2|PB|$. Determine the value of the real number $a$.

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