
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 parametric equation of curve $C_1$ is 
$$
\begin{cases} 
x=2\sqrt{5}\cos \alpha \\ 
y=2\sin \alpha
\end{cases}
(\alpha \text{ is the parameter}).
$$
In the polar coordinate system with the origin as the pole and the positive $x$-axis as the polar axis, the equation of curve $C_2$ is $\rho^2+4\rho\cos \theta-2\rho\sin \theta+4=0$.

(Ⅰ) Write the standard equations of curves $C_1$ and $C_2$;

(Ⅱ) A line $l$ passing through the left focus of curve $C_1$ and having a slope angle of $\frac{\pi}{4}$ intersects curve $C_2$ at points $A$ and $B$. Find $|AB|$.

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