
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 two sequences $\{a_n\}$ and $\{b_n\}$, with the first term $a_1=b_1=1$, and satisfying $(a_{n+1}-a_n)^2=4$, $|b_{n+1}|=q|b_n|$, where $n\in\mathbb{N}^*$. Let the sum of the first $n$ terms of $\{a_n\}$ and $\{b_n\}$ be $S_n$ and $T_n$ respectively.  
(I) If the inequality $a_{n+1} > a_n$ holds for all $n\in\mathbb{N}^*$, find $S_n$;  
(II) If the constant $q > 1$ and for any $n\in\mathbb{N}^*$, it always holds that $\sum\limits_{k=1}^{n+1}|b_k|\leqslant 4|b_n|$, find the value of $q$;  
(III) Under the condition of (II) and also satisfying the following two conditions:  
(i) If there exists a unique positive integer $p$ such that $a_p < a_{p-1}$;  
(ii) $T_m > 0$ always holds.  
Is there a positive integer $m$ such that $S_{m+1}=4b_m$? If it exists, find the value of $m$; if not, please explain why.

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