
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 the point $(1, \frac{1}{3})$ lies on the graph of the function $f(x)=a^{x}$ ($a > 0$ and $a \neq 1$), and the sum of the first $n$ terms of the geometric sequence $\{a_n\}$ is $f(n)-c$, the first term and the sum $S_n$ of the sequence $\{b_n\}$ ($b_n > 0$) satisfy $S_n-S_{n-1}= \sqrt{S_n}+ \sqrt{S_{n+1}}$ ($n \geqslant 2$).  
(1) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.  
(2) If the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{b_n b_{n+1}} \right\}$ is $T_n$, what is the smallest positive integer $n$ for which $T_n > \frac{1000}{2009}$?

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