
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
The general term of the sequence $\{a\_n\}$ is given by $a\_n=pn+q (n∈N^⁎, P > 0).$ The sequence $\{b\_n\}$ is defined as follows: For a positive integer $m$, $b\_m$ is the smallest value of $n$ that satisfies the inequality $a\_n≥m$.
(I) If $p= \frac {1}{2},q=- \frac {2}{3}$, find $b\_3$;
(II) If $p=2$, $q=-1$, find the sum of the first $2m$ terms of the sequence $\{b\_m\}$;
(III) Determine whether there exist $p$ and $q$ such that $b\_m=4m+1 (m∈N^⁎)$. If they exist, find the range of values for $p$ and $q$; if not, explain the reason.

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