
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 $a \in \mathbb{R}$, the function is defined as $f(x) = \log_2(\frac{1}{x} + a)$.
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, find the value of $a$.
3. Let $a > 0$. If for any $t \in [\frac{1}{2}, 1]$, the difference between the maximum and minimum values of the function $f(x)$ on the interval $[t, t+1]$ does not exceed 1, determine the range of values for $a$.

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