
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 a sequence {a<sub>n</sub>} with 16 terms, where a<sub>1</sub> = 1 and a<sub>8</sub> = 4. Define the function f<sub>n</sub>(x) as follows:

$$f_{n}(x)= \frac {1}{3}x^{3}-a_{n}x^{2}+(a_{n}^{2}-1)x$$

where n ∈ N*. If x = a<sub>n+1</sub> (1 ≤ n ≤ 15) is an extreme point of the function f<sub>n</sub>(x), and the slope of the tangent line to the curve y = f<sub>8</sub>(x) at the point (a<sub>16</sub>, f<sub>8</sub>(a<sub>16</sub>)) is 15, then the number of sequences {a<sub>n</sub>} that satisfy these conditions is \_\_\_\_\_\_.

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