
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 the arithmetic sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n= \frac{n^2+n}{2}$, and the geometric sequence $\{b_n\}$ with the sum of the first $n$ terms denoted as $T_n$, given $b_1=a_1+1$ and $b_2-a_2=2$,  
(1) Find the general term formulas for the sequences $\{a_n\}$ and $\{b_n\}$.  
(2) Find the smallest value of $n$ such that $T_n+a_n > 300$.

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