
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
Transform the second equation:
\[ y^{2}+2xy+x^{2}-6y-6x+5=0 \Leftrightarrow (y+x)^{2}-6(y+x)+5=0 \Leftrightarrow (y+x)^{2}-(y+x)-5(y+x)+5=0 \Leftrightarrow (y+x) \cdot (y+x-1)-5(y+x-1)=0 \Leftrightarrow (y+x-1) \cdot (y+x-5)=0 \]
Thus, we have:
\[ \begin{cases} y=1-x \\ y=5-x \end{cases} \]

Transform the first equation: 
\[ \left\{\begin{array}{c} y-x+1 = x^{2}-3x \\ x \neq 0 \\ x \neq 3 \end{array} \Leftrightarrow \left\{\begin{array}{c} y = x^{2}-2x-1 \\ x \neq 0 \\ x \neq 3 \end{array} \right. \right. \]

The last pair does not satisfy the condition \( x \neq 3 \).

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