
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
Suppose $a\in\mathbb{R}$, and for $x>0$, the function $f(x)$ is defined as $f(x)=\log_2\left(\frac{1}{x}+a\right)$.
(1) If the function $f(x)$ passes through the point $(1,1)$, find the explicit formula of $f(x)$ at this condition;
(2) If the function $g(x)=f(x)+2\log_2x$ has only one zero, find the value of the real number $a$;
(3) Let $a>0$. If for any real number $t \in \left[\frac{1}{3}, 1\right]$, the difference between the maximum and minimum values of $f(x)$ on the interval $[t, t+1]$ is no greater than $1$, determine the range of values for the real number $a$.

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