
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
A polynomial \( f \in \mathbb{Z}[x] \) is called splitty if and only if for every prime \( p \), there exist polynomials \( g_{p}, h_{p} \in \mathbb{Z}[x] \) with \( \operatorname{deg} g_{p}, \operatorname{deg} h_{p} < \operatorname{deg} f \) and all coefficients of \( f - g_{p} h_{p} \) are divisible by \( p \). Compute the sum of all positive integers \( n \leq 100 \) such that the polynomial \( x^{4} + 16 x^{2} + n \) is splitty.

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