
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 the quaterns  $(x,y,z,w)$  of real numbers (not necessarily distinct) that solve the following system of equations:  $$ x+y=z^2+w^2+6zw $$    $$ x+z=y^2+w^2+6yw $$    $$ x+w=y^2+z^2+6yz $$    $$ y+z=x^2+w^2+6xw $$    $$ y+w=x^2+z^2+6xz $$    $$ z+w=x^2+y^2+6xy $$  

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