
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 arithmetic sequence $\{a\_n\}$, it is known that $a\_1+a\_4+a\_7=9$, $a\_3+a\_6+a\_9=21$,
(I) Find the general term $a\_n$ of the sequence;
(II) Find the sum of the first 9 terms $S\_9$ of the sequence;
(III) If $c\_n=2^{a\_n+3}$, find the sum of the first $n$ terms $T\_n$ of the sequence $\{c\_n\}$.

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