
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 100 distinct cards with numbers \(2, 5, 2^{2}, 5^{2}, \ldots, 2^{50}, 5^{50}\) (each card has exactly one number, and each number occurs exactly once). In how many ways can you choose 2 cards such that the product of the numbers on the chosen cards is a cube of an integer?

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