
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 a function \( g(x) = \cos x \) where the vertical coordinates of all points on its graph are stretched to twice their original values (the horizontal coordinates remain unchanged), and the resulting graph is then shifted to the right by \( \frac{\pi}{2} \) units to obtain the function \( y = f(x) \). The equation \( f(x) + g(x) = m \) has two different solutions \( \alpha \) and \( \beta \) in the interval \([0, 2\pi)\).
(1) Find the range of the real number \( m \) that satisfies the given conditions.
(2) Determine \( \cos(\alpha - \beta) \) expressed in terms of \( m \).

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