
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
Let \( x, y, z \) be relative integers such that for any triangle \( ABC \) we have \( 16 HN^2 = xBC^2 + yCA^2 + zAB^2 \), where \( H \) is the foot of the altitude from \( A \), \( M \) is the midpoint of \( [BC] \), and \( N \) is the midpoint of \( [AM] \). Determine \( x^2 + y^2 + z^2 \).

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