
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 that $f(x)$ and $g(x)$ are odd and even functions defined on $R$ respectively, and $f(x) + g(x) = (\frac{1}{2})^{x}$. If there exists $x_{0} \in [\frac{1}{2}, 1]$ such that the equation $a f(x_{0}) + g(2 x_{0}) = 0$ holds true, then the range of values for the real number $a$ is _____.

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