
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 the function $f(x)=\lg \left(x+ \frac{a}{x}-2\right)$, where $a$ is a constant greater than $0$.

$(1)$ When $a \in (1,4)$, find the minimum value of the function $f(x)$ on the interval $[2,+\infty)$;

$(2)$ If for any $x \in [2,+\infty)$, $f(x) > 0$ always holds, determine the range of values for $a$.

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