
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
Compute the number of sequences of numbers \( a_{1}, a_{2}, \ldots, a_{10} \) such that:
I. \( a_{i}=0 \) or 1 for all \( i \),
II. \( a_{i} \cdot a_{i+1}=0 \) for \( i=1,2, \ldots, 9 \),
III. \( a_{i} \cdot a_{i+2}=0 \) for \( i=1,2, \ldots, 8 \).

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