
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 \(\log_{1+x^{2}}(1+27x^{5}) + \log_{1-2x^{2}+27x^{4}}(1+x^{2}) \leq 1 + \log_{1-2x^{2}+27x^{4}}(1+27x^{5})\). The solution set is \(x \in \left(-\frac{1}{\sqrt[5]{27}} , -\frac{1}{3}\right] \cup \left(-\sqrt{\frac{2}{27}} , 0\right) \cup \left(0 , \sqrt{\frac{2}{27}}\right) \cup \left\{\frac{1}{3}\right\}\).

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