
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
The first term of the arithmetic sequence $\{b_n\}$ is 1, and the common difference is 2. The sequence $\{a_n\}$ satisfies the relation $$b_{n}= \frac {a_{1}+2a_{2}+3a_{3}+\cdots +na_{n}}{1+2+3+\cdots +n}$$ for $n\in\mathbb{N}^*$. An odd function $f(x)$ is defined on $\mathbb{R}$, and for $x<0$, $$f(x)=- \frac {3^{qx}}{3^{qx}+p-1}.$$  
(1) Find the analytic expression of the function $f(x)$.  
(2) Find the general formula for the sequence $\{a_n\}$.  
(3) If $$\lim_{n\rightarrow \infty }f(a_{n})=0$$, find the condition that $p+q$ must satisfy.

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