
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 \( f(x) \) and \( g(x) \) be an odd function and an even function, respectively, defined on \( R \), such that \( f(x) + g(x) = 2^x \). If for \( x \in \left[\frac{1}{2}, 2\right] \), the inequality \( a f(x) - f(3x) \leq 2 g(2x) \) always holds, find the range of the real number \( a \).

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