
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 $a \in \mathbb{R}$, and the function $f(x) = \log_{2}\left(\frac{1}{x} + a\right)$

(1) When $a = 1$, solve the inequality $f(x) > 1$;

(2) If the equation $f(x) + \log_{2}(x^{2}) = 0$ has exactly one solution for $x$, find the value of $a$;

(3) If $a > 0$, and for any $t \in \left[\frac{1}{2}, 1\right]$, the difference between the maximum and minimum values of the function $f(x)$ on the interval $[t, t + 1]$ does not exceed $1$, find the range of values for $a$.

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