
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 respectively odd and even functions defined on $\mathbb{R}$, and $$f(x)+g(x)=\left( \frac{1}{2} \right)^{x}.$$  
(1) Find the explicit expressions for functions $f(x)$ and $g(x)$.  
(2) If there exists $x_0 \in \left[\frac{1}{2}, 1\right]$ such that the equation $af(x_0)+g(2x_0)=0$ holds, determine the range of the real number $a$.

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