
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
It is known that the numbers \(x, y, z\) form an arithmetic progression in the given order with a common difference \(\alpha = \arccos \frac{5}{9}\), and the numbers \(1 + \cos x, 1 + \cos y, 1 + \cos z\) form a non-constant geometric progression in the given order. Find \(\cos y\).

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