
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 $a \neq 1$, $m$, and $n$ are both positive numbers, then the minimum value of $\dfrac{1}{m+1}+ \dfrac{1}{2n}$ is \_\_\_\_\_\_.

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