
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 function $y = \log_a (x - m - 2n) + 2$ always passes through the fixed point $(3, 2)$, where $a > 0$ and $a \neq 1$, and both $m$ and $n$ are positive numbers, the minimum value of $\frac{1}{m + 1} + \frac{1}{2n}$ is __________.

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