
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 the ellipse $4x^{2}+y^{2}=1$ and the line $l$: $y=x+m$.
(1) For what values of $m$ does the line $l$ intersect the ellipse?
(2) If the length of the chord intercepted by the line $l$ on the ellipse is $\frac{4\sqrt{2}}{5}$, find the equation of the line $l$.

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