
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 the functions $f(x)=a^{x}$ and $g(x)=a^{2x}+m$, where $m > 0$, $a > 0$, and $a \neq 1$. When $x \in [-1, 1]$, the sum of the maximum and minimum values of $y = f(x)$ is $\frac{5}{2}$.
(I) Find the value of $a$;
(II) If $a > 1$, let the function $h(x) = g(x) - 2mf(x)$. Find the minimum value $H(m)$ of $h(x)$ when $x \in [0, 1]$;
(III) If $a > 1$, and the inequality $|\frac{f(x) - mg(x)}{f(x)}| \leq 1$ always holds when $x \in [0, 1]$, find the range of $m$.

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