
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 sequence $\{a_n\}$, the sum of the first $n$ terms $S_n = \frac{n^2+n}{2}$, and another sequence $\{b_n\}$, where the nth term is denoted by $b_n = f(n)$ and $f(n)$ satisfies: ① $f(1) = \frac{1}{2}$; ② For any positive integers $m$ and $n$, $f(m+n) = f(m)f(n)$ holds.  
(1) Find $a_n$ and $b_n$;  
(2) Let the sum of the first $n$ terms of the sequence $\{a_nb_n\}$ be $T_n$, find $T_n$.

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