
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 \( n \) be the positive integer such that
\[ \frac{1}{9 \sqrt{11} + 11 \sqrt{9}} + \frac{1}{11 \sqrt{13} + 13 \sqrt{11}} + \frac{1}{13 \sqrt{15} + 15 \sqrt{13}} + \cdots + \frac{1}{n \sqrt{n+2} + (n+2) \sqrt{n}} = \frac{1}{9} . \]

Find the value of \( n \).

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