
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 $f(x)$ is an even function and $g(x)$ is an odd function on $\mathbb{R}$, and $f(x) + g(x) = \log_{4}(4^x + 1)$.

1. Find the analytical expressions for $f(x)$ and $g(x)$.
2. If the function $h(x) = f(x) - \frac{1}{2} \log_{2}(a \cdot 2^x + 2\sqrt{2}a)$, where $a > 0$, has only one zero on $\mathbb{R}$, find the range of values for the real number $a$.

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