
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
An ellipse has the equation $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$ with a major axis length of 4 and a focal distance of 2.
(1) Find the standard equation of the ellipse;
(2) A line $l$ passing through the left focus $F$ of the ellipse at an angle of $\frac{\pi }{4}$ intersects the ellipse at points $A$ and $B$. Find the length of the chord $|AB|$.

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