
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 $\overrightarrow{a}=(1,2)$ and $\overrightarrow{b}=(-2,3)$, if $m\overrightarrow{a}-n\overrightarrow{b}$ is collinear with $\overrightarrow{a}+2\overrightarrow{b}$ (where $m,n\in\mathbb{R}$ and $n\neq 0$), find the value of $\frac{m}{n}$.

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