
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
Given an ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1$ ($a > b > 0$) with an eccentricity of $\frac{\sqrt{5}}{5}$, and its left focus is at $F(-1,0)$.

(1) Find the equation of ellipse $C$;
(2) If a line $l$ passes through point $F$ with a slope of $\frac{\pi}{4}$ and intersects the ellipse $C$ at points $A$ and $B$, find the length of chord $|AB|$.

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