
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
Given $f(x)$ and $g(x)$ be odd and even functions defined on $\mathbb{R}$, respectively, and $f(x) + g(x) = 2^x$. If the inequality $a \cdot f(x) + g(2x) \geq 0$ holds for all $x \in [1, 2]$, determine the range of the real number $a$.

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