
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 a sequence of $m$ terms ($m \in \mathbb{N}^*$, $m \geq 3$), denoted as $\{a_n\}$, where $a_i \in \{0, 1\}$ ($i=1, 2, \ldots, m$). If there exists a positive integer $k$ ($2 \leq k \leq m-1$) such that there are two sets of consecutive $k$ terms in the sequence $\{a_n\}$ that exactly match each other in order, then the sequence $\{a_n\}$ is called a "kth-order repeatable sequence". For example, for the sequence $\{a_n\}$: 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 each other in order, the sequence $\{a_n\}$ is a "4th-order repeatable sequence".
- (Ⅰ) Determine whether the following sequences are "5th-order repeatable sequences". If yes, please write down the repeating 5 terms;
  ① $\{b_n\}$: 0, 0, 0, 1, 1, 0, 0, 1, 1, 0.
  ② $\{c_n\}$: 1, 1, 1, 1, 1, 0, 1, 1, 1, 1.
- (Ⅱ) If a sequence of $m$ terms $\{a_n\}$ is definitely a "3rd-order repeatable sequence", what is the minimum value of $m$? Explain your reasoning;
- (Ⅲ) Assuming the sequence $\{a_n\}$ is not a "5th-order repeatable sequence", if adding one more term (either 0 or 1) after the last term $a_m$ makes the new sequence a "5th-order repeatable sequence", and given $a_4 \neq 1$, find the value of the last term $a_m$.

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