
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

Consider the ellipse \(\frac{x^{2}}{2} + y^{2} = 1\) with foci \(F_{1}\) and \(F_{2}\). A line \(l\), which does not pass through the focus \(F_{1}\), intersects the ellipse at two distinct points \(A\) and \(B\). The distance from the focus \(F_{2}\) to the line \(l\) is \(d\). If the slopes of \(A F_{1}\), \(l\), and \(B F_{1}\) form an arithmetic sequence, determine the possible range of values for \(d\).

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