
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
There are some natural numbers (excluding zero) that are both perfect squares and perfect cubes. (Note: A perfect square can be written as the product of two identical natural numbers, and a perfect cube can be written as the product of three identical natural numbers). For example: \(1 = 1 \times 1 = 1 \times 1 \times 1\) and \(64 = 8 \times 8 = 4 \times 4 \times 4\). How many such numbers are there within 1000?

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