
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
From the sequence \( x_{1}, x_{2}, \ldots, x_{n} \) where \( x_{i} \in \{0,1\} \), a new sequence \( y_{1}, y_{2}, \ldots, y_{n-1} \) is derived using the rule: \( y_{i} = x_{i} \cdot x_{i+1} \) for \( i = 1, \ldots, n-1 \). Determine which of the four sequences below \( y_{1}, y_{2}, \ldots, y_{10} \) could have been obtained in this manner, and which could not.

(I): 0011001100  
(II): 0001111101  
(III): 1000111000  
(IV): 1100110110

How many sequences \( y_{1}, y_{2}, \ldots, y_{6} \) can be obtained (considering all possible choices of \( x_{1}, x_{2}, \ldots, x_{7} \) where \( x_{i} \in \{0,1\} \))? Justify your answer.

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