
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, the curve defined by $$\begin{cases} x=4\cos\alpha \\ y=\sin\alpha\end{cases}$$ (where $\alpha$ is a parameter) undergoes the following transformations: each point's $y$-coordinate remains unchanged, the $x$-coordinate is halved, then the entire graph is shifted right by 1 unit, and finally, the $y$-coordinate is doubled while the $x$-coordinate remains unchanged, resulting in curve $C_1$. In a polar coordinate system with the origin as the pole and the non-negative $x$-axis as the polar axis, the equation of curve $C_2$ is $\rho=4\sin\theta$. Find the length of the common chord of $C_1$ and $C_2$.

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