
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
Let $F_1$ and $F_2$ be the left and right foci of the ellipse $E$: $x^{2}+ \frac {y^{2}}{b^{2}}=1(0 < b < 1)$, respectively. A line $l$ passing through $F_1$ intersects $E$ at points $A$ and $B$, and $|AF_{2}|$, $|AB|$, $|BF_{2}|$ form an arithmetic sequence.  
$(1)$ Find $|AB|$;  
$(2)$ If the slope of line $l$ is $1$, find the value of the real number $b$.

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