
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 $C$ with a major axis length of $2\sqrt{2}$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ and one focus at $(-1,0)$.<br/>$(Ⅰ)$ Find the equation of the ellipse $C$;<br/>$(Ⅱ)$ If a line $l$ with a slope of $1$ intersects the ellipse $C$ at points $A$ and $B$, and $|AB|=\frac{4\sqrt{2}}{3}$, find the equation of the line $l$.

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