
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 that the right focus of an ellipse is $F(m,0)$, the left and right directrices are $l_{1}$: $x=-m-1$, $l_{2}$: $x=m+1$, respectively. The lines $l_{1}$ and $l_{2}$ intersect with the line $y=x$ at points $A$ and $B$, respectively.
1. If the eccentricity is $\frac{\sqrt{2}}{2}$, find the equation of the ellipse.
2. When $\overrightarrow{AF} \cdot \overrightarrow{FB} < 7$, find the range of the eccentricity.

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