
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
If \( d \) is a common divisor of the differences \( a_{1} - a_{2} \), \( b_{1} - b_{2} \), and \( c_{1} - c_{2} \), then \( a_{1} b_{1} c_{1} - a_{2} b_{2} c_{2} \) is also divisible by \( d \).

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