
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
Let $f(x), g(x), T_{1}, T_{2}$ be as defined in Theorem 2. If $f(x)$ and $g(x)$ always take positive values, then a necessary and sufficient condition for $f(x) g(x)$ and $f(x) / g(x)$ to be periodic functions is that $T_{1}$ and $T_{2}$ are commensurable.

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