
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
Positive rational numbers  $x<y<z$  sum to  $1$  and satisfy the equation  $$ (x^2+y^2+z^2-1)^3+8xyz=0. $$  Given that  $\sqrt{z}$  is also rational, it can be expressed as  $\frac{m}{n}$  for relatively prime positive integers  $m$  and  $n$ . If  $m+n < 1000$ , find the maximum value of  $m+n$ .

*Proposed by **Th3Numb3rThr33***

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