
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 distinct natural numbers \( k, l, m, n \), it is known that there exist three natural numbers \( a, b, c \) such that each of the numbers \( k, l, m, n \) is a root of either the equation \( a x^{2} - b x + c = 0 \) or the equation \( c x^{2} - 16 b x + 256 a = 0 \). Find \( k^{2} + l^{2} + m^{2} + n^{2} \).

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