
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
Find the number of permutations  $x_1, x_2, x_3, x_4, x_5$  of numbers  $1, 2, 3, 4, 5$  such that the sum of five products  $$ x_1x_2x_3 + x_2x_3x_4 + x_3x_4x_5 + x_4x_5x_1 + x_5x_1x_2 $$  is divisible by  $3$ .

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