
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
Highlight the solution \( (0,0) \) by dividing both parts of the first equation by \( xy \) and the second by \( x^{2} y^{2} \). We get:

\[
\left\{\begin{array}{c}\left(\frac{x}{y}+\frac{y}{x}\right)(x+y)=15 \\ \left(\frac{x^{2}}{y^{2}}+\frac{y^{2}}{x^{2}}\right)\left(x^{2}+y^{2}\right)=85\end{array}\right.
\]

Let \( x + y = a \ and \ \frac{x}{y} + \frac{y}{x} = b \), then the first equation will be:

\[
ab = 15
\]

Considering that:

\[
\frac{x^{2}}{y^{2}} + \frac{y^{2}}{x^{2}} = b^{2} - 2 \ and \ xy = \frac{a^{2}}{2+b}
\]

Then, the second equation can be written as:

\[
(b^{2} - 2) a^{2} b = 85(2 + b)
\]

Substitute the expression for \( a \) from the first equation into this, we get:

\[
\frac{3}{b}(b^{2} - 2) 15 = 17(2 + b) \Rightarrow 14b^{2} - 17b - 45 = 0
\]

Then:

\[
b_{1} = -\frac{9}{7}, \ b_{2} = \frac{5}{2}
\]

Then:

\[
a_{1} = \frac{15}{b} = -\frac{15}{\frac{9}{7}} = -\frac{35}{3}, \ a_{2} = 6
\]

We have two systems:

1) \(\left\{\begin{array}{l}x + y = 6 \\ \frac{x}{y} + \frac{y}{x} = \frac{5}{2}\end{array}\right.\)

2) \(\left\{\begin{array}{l}x + y = -\frac{35}{3} \\ \frac{x}{y} + \frac{y}{x} = -\frac{9}{7}\end{array}\right.\)

Solve the first system:

\(\left\{\begin{array}{c}x + y = 6 \\ x^{2} + y^{2} = \frac{5}{2} xy\end{array} \Rightarrow \left\{\begin{array}{c} x + y = 6 \\ (x + y)^{2} = 4.5xy\end{array}\Rightarrow \left\{\begin{array}{c}x + y = 6 \\ 4.5xy = 36\end{array}\right.\right.\right.\)

\(\left\{\begin{array}{c}x + y = 6 \\ xy = 8\end{array}\right.\)

By Vieta's theorem:
\[
x = 2, \ y = 4 \ or \ x = 4, \ y = 2
\]

Solve the second system:

\(\left\{\begin{array}{l}x + y = -\frac{35}{3} \\ \frac{x}{y} + \frac{y}{x} + \frac{9}{7} = 0\ \left(\frac{y}{x}\right)^{2}+\frac{9}{7}\left(\frac{y}{x}\right)+1=0 \quad D<0\right.\}\) has no solutions.

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