
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
For any \( x \in \mathbf{R} \), \( f(x) = |\sin x| \). When \( n \leq x < n+1 \) ( \( n \) is an integer), \( g(x) = x - n \). How many of the functions \( f(x) \), \( g(x) \), \( f(x) + g(x) \), and \( f(x) g(x) \) are periodic functions?

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