
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
Given a positive geometric sequence $\{{{a_{n}}}\}$ that satisfies ${{a_{4}}}={{a_{2}}} \cdot {{a_{3}}}$, and the sum of the first three terms ${{S_{3}}=13}$.

(1) Find ${{a_{n}}}$;

(2) If sequence $\{{{b_{n}}}\}$ satisfies ${{b_{n}}={\log _{3}}{{a_{n}}+n}}$, and the sum of the first $n$ terms of the sequence $\{\frac{1}{{{b_{n}}{{b_{n+1}}}}}\}$ is ${{T_{n}}}$, find ${{T_{n}}}$.

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