
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: 
\[ y^{2}+2xy+x^{2}-6y-6x+5=0 \]
which can be rewritten as:
\[ (y+x)^{2}-6(y+x)+5=0 \]
\[ (y+x)^{2}-(y+x)-5(y+x)+5=0 \]
\[ (y+x) \cdot (y+x-1)-5(y+x-1)=0 \]
\[ (y+x-1) \cdot (y+x-5)=0 \]
Hence, the solutions are:
\[ y = 1-x \]
\[ y = 5-x \]

Now, transform the first equation:
\[ y - x + 1 = x^{2} - 3x \]
along with the conditions:
\[ x \neq 0 \]
\[ x \neq 3 \]
which simplifies to:
\[ y = x^{2} - 2x - 1 \]
with the same conditions:
\[ x \neq 0 \]
\[ x \neq 3 \]

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