
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 functions defined on \(\mathbf{R}\), with \( f(x) \) being an odd function and \( g(x) \) being an even function, such that \( f(x) + g(x) = 2^x \). If the inequality \( a f(x) + g(2x) \geqslant 0 \) holds for all \( x \in [1, 2] \), determine the range of the real number \( a \).

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