
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 positive integer solutions  $(x,y,z,n)$  of equation  $x^{2n+1}-y^{2n+1}=xyz+2^{2n+1}$ , where  $n\ge 2$  and  $z \le 5\times 2^{2n}$ .

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