
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
Let $f(x)=\log_{2}(\frac{1}{x}+a)$, where $a\in \mathbb{R}$. <br/> $(1)$ When $a=2$, find the solution set of the inequality $f(x) < 2$; <br/> $(2)$ When $a > 0$, 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 $1$, find the range of values for $a$.

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