
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]


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Given that a positive integer \( A \) can be factorized as \( A = 2^{\alpha} \times 3^{\beta} \times 5^{\gamma} \), where \( \alpha \), \( \beta \), and \( \gamma \) are natural numbers, and given that half of \( A \) is a perfect square, one-third of \( A \) is a perfect cube, and one-fifth of \( A \) is a perfect fifth power of some natural number, determine the minimum value of \( \alpha + \beta + \gamma \).

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