
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
Let $\{a\_n\}$ be an increasing geometric sequence, given that $a\_1 + a\_3 = 5$, and $a\_1 + 3$, $3a\_2$, $a\_3 + 4$ form an arithmetic sequence.

1. Find the general term of the sequence $\{a\_n\}$.
2. Define a new sequence $\{b\_n = \ln a_{3n+1}}\}$, where $n=1, 2, ...$, find the sum of the first $n$ terms, denoted as $T\_n$.

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