
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

(a) Determine a real number \(w\) for which \(\frac{1}{w}, \frac{1}{2}, \frac{1}{3}, \frac{1}{6}\) is an arithmetic sequence.

(b) Suppose \(y, 1, z\) is a geometric sequence with \(y\) and \(z\) both positive. Determine all real numbers \(x\) for which \(\frac{1}{y+1}, x, \frac{1}{z+1}\) is an arithmetic sequence for all such \(y\) and \(z\).

(c) Suppose that \(a, b, c, d\) is a geometric sequence and \(\frac{1}{a}, \frac{1}{b}, \frac{1}{d}\) is an arithmetic sequence with each of \(a, b, c,\) and \(d\) positive and \(a \neq b\). Determine all possible values of \(\frac{b}{a}\).

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