
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
For the sequence $A$: $a_{1}$, $a_{2}$, $…$, $a_{n}$, if it satisfies that $a_{i} \in \{0,1\}$ for $i=1, 2, 3, …, n$, then the sequence $A$ is called a "0-1 sequence". If there exists a positive integer $k$ ($2 \leq k \leq n-1$) such that there are two continuous subsequences of length $k$ within the sequence $\{a_{n}\}$ which exactly match each other in order, then the sequence $\{a_{n}\}$ is called a "kth-order repeatable sequence". For example, the sequence $A$: $0, 1, 1, 0, 1, 1, 0$ is a "4th-order repeatable sequence" since the subsequence $a_{1}, a_{2}, a_{3}, a_{4}$ matches the subsequence $a_{4}, a_{5}, a_{6}, a_{7}$ in order.
- (I) Determine whether the following sequence $A$: $1, 1, 0, 1, 0, 1, 0, 1, 1, 1$, is a "5th-order repeatable sequence"? If it is, please write out the repeating subsequence of length $5$.
- (II) If a sequence $A$ of length $m$ is a "3rd-order repeatable sequence", what is the minimum value of $m$? Explain the reason.
- (III) Assuming the sequence $A$ is not a "5th-order repeatable sequence", if one more term $0$ or $1$ is added after the last term $a_{m}$, the new sequence becomes a "5th-order repeatable sequence", and $a_{4}=1$, find the value of the last term $a_{m}$.

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