
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 is an expression written on the board, $\frac{a}{b} \cdot \frac{c}{d} \cdot \frac{e}{f}$, where $a, b, c, d, e, f$ are natural numbers. If the number $a$ is increased by 1, the value of this expression increases by 3. If the number $c$ in the original expression is increased by 1, its value increases by 4. If the number $e$ in the original expression is increased by 1, its value increases by 5. What is the smallest value that the product $b d f$ can take?

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