
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 that the terms of the sequence $\{a_n\}$ are all non-negative integers, and satisfy $a_1=0$, $a_2=3$, $a_{n+1}a_n=(a_{n-1}+2)(a_{n-2}+2)$ $(n\in\mathbb{N}^*, n\geqslant 3)$, calculate the value of $a_9+a_{10}$.

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