
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 $a \in \mathbb{R}$, when $x > 0$, $f(x) = \log_{2}\left(\frac{1}{x} + a\right)$.
(1) If the function $f(x)$ passes through the point $(1, 1)$, find the expression of the function $f(x)$;
(2) If the function $g(x) = f(x) + 2\log_{2}{x}$ has only one zero point, find the value of the real number $a$;
(3) Given that $a > 0$, if for any real number $t \in \left[\frac{1}{3}, 1\right]$, the difference between the maximum and minimum values of the function $f(x)$ on $\left[t, t+1\right]$ is not greater than $1$, find the range of values of the real number $a$.

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