
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
Let the sequence $\{a_n\}$ be defined by the general term $a_n = pn + q$ ($n \in \mathbb{N}^*$, $P > 0$). The sequence $\{b_n\}$ is defined as follows: for a positive integer $m$, $b_m$ is the smallest value of $n$ such that the inequality $a_n \geq m$ holds.

(Ⅰ) If $p= \frac{1}{2}$ and $q= -\frac{2}{3}$, find $b_3$.

(Ⅱ) If $p=2$ and $q=-1$, find the formula for the sum of the first $2m$ terms of the sequence $\{b_m\}$.

(Ⅲ) Is there any $p$ and $q$ such that $b_m = 4m + 1$ ($m \in \mathbb{N}^*$)? If yes, find the range of values for $p$ and $q$; if no, explain why.

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