
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
Given two quadratic functions \( f_{1}(x) = a_{1} x^{2} + b_{1} x + c_{1} \) and \( f_{2}(x) = a_{2} x^{2} + b_{2} x + c_{2} \), such that \( f_{1}(x) - f_{2}(x) \) is a monotonically decreasing function on the interval \([1, 2]\), and has a maximum value of 5 and a minimum value of 3 on this interval, find a pair of functions \( f_{1}(x) \) and \( f_{2}(x) \) that satisfy these conditions.

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