
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 an arithmetic sequence $\{a\_n\}$, it is known that $a\_2+a\_4=10$ and $a\_5=9$. Another sequence $\{b\_n\}$ is defined by $b\_1=a\_1$ and $b\_{n+1}=b\_n+a\_n$.

1. Find the general term formula for the sequence $\{a\_n\}$;
2. If $c\_n=\frac{1}{a\_n \cdot a\_{n+1}}$, find the sum of the first $n$ terms $T\_n$ of the sequence $\{c\_n\}$;
3. Find the general term formula for the sequence $\{b\_n\}$.

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