
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 $F_1$ and $F_2$ are the left and right foci of the ellipse $E: x^2 + \frac{y^2}{b^2} = 1$ $(0 < b < 1)$, and the line $l$ passing through $F_1$ intersects $E$ at points $A$ and $B$. If the distances $|AF_2|$, $|AB|$, and $|BF_2|$ form an arithmetic sequence:

(1) Find $|AB|$;

(2) If the slope of line $l$ is $1$, find the value of $b$.

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