
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 sequence $\{a_{n}\}$ such that $a_{1}=\frac{1}{2}$ and $a_{n}=2a_{n}a_{n+1}+3a_{n+1}$ for all natural numbers $n$:

1. Find the general term of the sequence $\{a_{n}\}$.
2. If the sequence $\{b_{n}\}$ satisfies $b_{n}=1+\frac{1}{a_{n}}$ for all natural numbers $n$, and the inequality $\sum_{k=1}^{n} \frac{1}{n+\log_{3} b_{k}}>\frac{m}{24}$ holds for any positive integer $n \geq 2$, find the maximum integer value of $m$.

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