
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 $C_{1}$ is given by $\begin{cases} x=t+1 \\ y=1-2t\end{cases}$ (where $t$ is a parameter) and the curve $C_{2}$ is given by $\begin{cases} x=a\cos \theta \\ y=3\sin \theta\end{cases}$ (where $\theta$ is a parameter, $a > 0$).

(Ⅰ) If curve $C_{1}$ and curve $C_{2}$ have one common point on the $x$-axis, find the value of $a$;

(Ⅱ) When $a=3$, curves $C_{1}$ and $C_{2}$ intersect at points $A$ and $B$. Find the distance between points $A$ and $B$.

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