
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 rectangular coordinate system $(xOy)$, the parametric equations of the curve $C\,_1$ are given by $\begin{cases} x=2\sqrt{5}\cos \alpha \\ y=2\sin \alpha \end{cases}$ ($\alpha$ is the parameter). In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the curve $C\,_2$ is defined by the equation $\rho\,^2+4\rho\cos \theta-2\rho\sin \theta+4=0$.
(I) Write the Cartesian equations of the curves $C_1$ and $C_2$;
(II) The line $l$ passing through the left focus of the curve $C_1$ with a slope angle of $\frac{\pi }{4}$ intersects the curve $C_2$ at points $A$ and $B$. Calculate $|AB|$.

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