
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 a focal length of $4 \sqrt {2}$, and a short semi-axis length of $2$, a line $l$ passes through point $P(-2,1)$ with a slope of $1$ and intersects the ellipse $C$ at points $A$ and $B$.

(1) Find the standard equation of the ellipse $C$.

(2) Find the length of the chord $AB$.

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