
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 $a \in \mathbb{R}$, the function is defined as $f(x) = \log_{2}\left(\frac{1}{x} + a\right)$.
(1) When $a = 5$, solve the inequality $f(x) > 0$;
(2) If the equation $f(x) - \log_{2}[(a - 4)x + 2a - 5] = 0$ has exactly one solution for $x$, find the range of $a$;
(3) Given that $a > 0$, if the difference between the maximum and minimum values of the function $f(x)$ on the interval $[t, t + 1]$ does not exceed $1$ for any $t \in [\frac{1}{2}, 1]$, find the range of $a$.

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