
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
In the infinite sequence $\{a\_n\}$, $a\_1=p$ is a positive integer, and satisfies $a_{n+1}= \begin{cases} \frac {a\_n}{2}, \text{when }a\_n\text{ is even} \\ a\_n+5, \text{when }a\_n\text{ is odd}. \end{cases}$

(I) Find the value of $p$ when $a_{3}=9$; (The conclusion does not require proof)

(II) Suppose $p=7$, and the sum of the first $n$ terms of the sequence $\{a\_n\}$ is $S\_n$, find $S_{150}$;

(III) If there exists $m∈N^{∗}$, such that $a_{m}=1$, find all values of $p$.

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