
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 that $$\sqrt {x}, \frac { \sqrt {f(x)}}{2}, \sqrt {3} (x \geq 0)$$ form an arithmetic sequence. Also, in the sequence $a_n$ ($a_n > 0$) with $a_1 = 3$, the sum of the first $n$ terms of this sequence $S_n$ ($n \in \mathbb{N}^+$) satisfies $S_n = f(S_{n-1})$ for all positive integers $n > 1$.  
(1) Find the $n+1$-th term of the sequence $a_n$;  
(2) If $\sqrt {b_{n}}$ is the geometric mean of $\frac {1}{a_{n+1}}$ and $\frac {1}{a_{n}}$, and $T_n$ is the sum of the first $n$ terms of $\{b_n\}$, find $T_n$.

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