
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
A sequence \(a_{1}, a_{2}, a_{3}, a_{4}, \cdots\), with \(a_{1}=\frac{1}{2}\), is defined by
\[ a_{n}=2 a_{n} a_{n+1}+3 a_{n+1} \]
for all \(n=1,2,3, \cdots\). If \(b_{n}=1+\frac{1}{a_{n}}\) for all \(n=1,2,3, \cdots\), find the largest integer \(m\) such that
\[ \sum_{k=1}^{n} \frac{1}{\log _{3} b_{k}}>\frac{m}{24} \]
for all positive integer \(n \geq 2\).

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