
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
Fill the numbers 1, 2, 3, 4, 5, 6, 7, 8, 9 into the circles in the diagram so that each circle contains exactly one number, satisfying the following conditions:
(1) The sum of the numbers on each side of the equilateral triangle is equal;
(2) The sum of the squares of the numbers on each side of the equilateral triangle leaves the same remainder when divided by 3.
Question: How many different ways are there to fill in the numbers?
(Note: Configurations that are identical through rotation and reflection are considered the same.)

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