
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
Solve the inequality

$$
\frac{\left(2 \cdot 3^{-\log _{x} 2}-6\right) \sqrt{2-\sqrt{2 \log _{x} 2+3}}}{2+\sqrt{\log _{x} 2+2}}>\frac{\left(3^{-\log _{x} 2}-3\right) \sqrt{2-\sqrt{2 \log _{x} 2+3}}}{\sqrt{\log _{x} 2+2}-1}
$$

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