
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 \(x^{2} \leq 2([\sqrt[3]{x}+0.5]+[\sqrt[3]{x}])\), where \([x]\) is the integer part of \(x\), i.e., \([x]\) is the integer for which the inequality \([x] \leq x < [x] + 1\) holds. Provide the difference between the largest and smallest solutions of the inequality.

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