
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
On the board, there is an expression \(\frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f}\), where \(a, b, c, d, e, f\) are natural numbers. If the number \(a\) is increased by 1, then the value of this expression increases by 3. If in the original expression the number \(c\) is increased by 1, its value increases by 4; if the number \(e\) is increased by 1, its value increases by 5. What is the smallest possible value of the product \(bdf\)?

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