
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 the function $f(x)$ is defined on the domain $\mathbb{R}$ and its graph passes through the point $(1,1)$. For any real numbers $x_1 < x_2$, we have $\frac{f(x_1)-f(x_2)}{x_1-x_2} > -2$. Determine the solution set of the inequality $f(\log_2|3x-1|) < 3 - \log_{\sqrt{2}}|3x-1|$.

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