
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]


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Given a sequence ${{a_{n}}}$ where all terms are non-zero, the sum of the first $n$ terms is ${{S_{n}}}$, and it satisfies ${{a_{1}}=a,}$ $2{{S_{n}}={{a_{n}}{{a_{n+1}}}}}$.

(I) Find the value of ${{a_{2}}}$;
(II) Find the general formula for the $n^{th}$ term of the sequence;
(III) If $a=-9$, find the minimum value of ${{S_{n}}}$.

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