
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
There are three positive numbers that may be inserted between $4$ and $16$ such that the sequence forms a harmonic progression (HP) for the first four numbers and the last four numbers form a quadratic sequence. Let the quadratic sequence be represented by $n^2 + pn + q$, where $n$ is the position in the sequence. Find the sum of those three positive numbers.
A) 25
B) 30
C) 33
D) 36

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