
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 $f(x)=\log\frac{{\lambda x+1}}{x}$.  
$(1)$ When $\lambda =2$, find the solution to the inequality $f(x) > 0$.  
$(2)$ Let $\lambda > 0$. For any $a \in [\frac{1}{2}, 2]$, and any $x_{1}, x_{2} \in [a, a+1]$, it holds that $|f(x_{1})-f(x_{2})| \leq \log 2$. Find the range of values for $\lambda$.

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