
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 $\overrightarrow{{e}_{1}}, \overrightarrow{{e}_{2}}$ be two non-collinear vectors in a plane, $\overrightarrow{AB}=(a-1) \overrightarrow{{e}_{1}}+ \overrightarrow{{e}_{2}}, \overrightarrow{AC}=b \overrightarrow{{e}_{1}}-2 \overrightarrow{{e}_{2}}$ where $a>0,b>0$. Find the minimum value of $\frac{1}{a}+ \frac{2}{b}$ given that points A, B, and C are collinear.

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