
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 integer solutions for the following equations:

1. $\frac{5}{x}-\frac{7}{y}=2$
2. $a^{4}+b^{4}+c^{4}+\cdots+m^{4}+n^{4}=1599$
3. $y^{4}=x^{3}+7$
4. $3 \times 2^{x}+1=y^{2}$
5. $a^{2}+b^{2}+c^{2}=a^{2} b^{2}$
6. $xy+yz+zx=xyz+2 \quad (x, y, z$ strictly positive$)$
7. $(xy-7)^{2}=x^{2}+y^{2}$
8. $3(xy+yz+zx)=4xyz \quad (x, y, z$ strictly positive$)$

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