
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
How many 6-tuples  $ (a_1,a_2,a_3,a_4,a_5,a_6)$  are there such that each of  $ a_1,a_2,a_3,a_4,a_5,a_6$  is from the set  $ \{1,2,3,4\}$  and the six expressions
\[ a_j^2 \minus{} a_ja_{j \plus{} 1} \plus{} a_{j \plus{} 1}^2\]
for  $ j \equal{} 1,2,3,4,5,6$  (where  $ a_7$  is to be taken as  $ a_1$ ) are all equal to one another?

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