
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
We call a positive integer \( t \) good if there is a sequence \( a_{0}, a_{1}, \ldots \) of positive integers satisfying \( a_{0}=15, a_{1}=t \), and
$$
a_{n-1} a_{n+1}=\left(a_{n}-1\right)\left(a_{n}+1\right)
$$
for all positive integers \( n \). Find the sum of all good numbers.

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