
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 vectors $\overrightarrow{m}=(1,a)$ and $\overrightarrow{n}=(2b-1,3)$ where $a>0$ and $b>0$. If $\overrightarrow{m} \cdot \overrightarrow{n}=1$, calculate the minimum value of $\frac{1}{a}+\frac{2}{b}$.

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