
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: $a_1, a_2, \ldots, a_n$, if it satisfies $a_i \in \{0,1\}$ ($i=1,2,3,\ldots,n$), then 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 sets of consecutive $k$ terms in sequence $\{a_n\}$ that exactly match in order, then sequence $\{a_n\}$ is called a "k-order repeatable sequence". For example, for sequence A: 0, 1, 1, 0, 1, 1, 0. Since $a_1, a_2, a_3, a_4$ and $a_4, a_5, a_6, a_7$ match in order, sequence $\{a_n\}$ is a "4-order repeatable sequence".
- (I) Determine whether the following sequence A: 1, 1, 0, 1, 0, 1, 0, 1, 1, 1. is a "5-order repeatable sequence"? If yes, please write down these 5 repeating terms.
- (II) If a sequence A with $m$ terms is definitely a "3-order repeatable sequence", what is the minimum value of $m$? Explain your reasoning.
- (III) Assuming sequence A is not a "5-order repeatable sequence", if adding one more term (either 0 or 1) after the last term $a_m$ makes the new sequence a "5-order repeatable sequence", and $a_4 \neq 1$, find the value of the last term $a_m$ of sequence $\{a_n\}$.

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