
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
Determine the necessary and sufficient condition under which the value of the fraction \(\frac{a x^{2} + b x + c}{a_{1} x^{2} + b_{1} x + c_{1}}\) remains unchanged for arbitrary values of the variable \(x\). Generalize this condition.

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