
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
The numbers $a_1, a_2, a_3, b_1, b_2, b_3, c_1, c_2, c_3$ are equal to the numbers $1, 2, 3, \dots, 9$ in some order. Find the smallest possible value of
\[
a_1 a_2 a_3 + b_1 b_2 b_3 + c_1 c_2 c_3
\]
given that each triplet $(a_1, a_2, a_3)$, $(b_1, b_2, b_3)$, $(c_1, c_2, c_3)$ must contain at least one even and one odd number.

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