
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 following inequality:

\[
\frac{\left(2 \cdot 2^{-\log_{x} 3} - 4\right) \sqrt{2 - \sqrt{\log_{x} 3 + 2}}}{1 + \sqrt{\log_{x} 3 + 5}} > \frac{\left(2^{-\log_{x} 3} - 2\right) \sqrt{2 - \sqrt{\log_{x} 3 + 2}}}{\sqrt{\log_{x} 3 + 5} - 2}
\]

By making the substitution \( y = \log_{x} 3 \), solve for \( x \).

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