
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 all tuples  $ (x_1,x_2,x_3,x_4,x_5,x_6)$  of non-negative integers, such that the following system of equations holds:
 $ x_1x_2(1\minus{}x_3)\equal{}x_4x_5 
x_2x_3(1\minus{}x_4)\equal{}x_5x_6 
x_3x_4(1\minus{}x_5)\equal{}x_6x_1 
x_4x_5(1\minus{}x_6)\equal{}x_1x_2 
x_5x_6(1\minus{}x_1)\equal{}x_2x_3 
x_6x_1(1\minus{}x_2)\equal{}x_3x_4$ 

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