
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 $a \in R$, the functions are defined as $f(x) = \log_{2}\left(\frac{1}{x} + a\right)$ and $g(x) = \log_{2}\left[(a - 4)x + 2a - 5\right]$.

(1) Solve the inequality $g(x) > 0$.
(2) If the graphs of the functions $y=f(x)$ and $y=g(x)$ have exactly one common point, find the range of values for $a$.
(3) Given that $a > 0$, if for any $t \in \left[\frac{1}{2}, 1\right]$, the function satisfies $\left|f(x_{1}) - f(x_{2})\right| \leq 1$ for any $x_{1}, x_{2} \in \left[t, t+1\right]$, find the range of values for $a$.

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