
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 a function defined on \( \mathbb{R} \), where \( f(x) \) is an odd function and \( g(x) \) is an even function. Given that \( f(x) + g(x) = 2^{x} \), for \( x \in [1,2] \), the inequality \( a f(x) + g(2x) \geq 0 \) holds. Determine the range of the real number \( a \).

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