
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
We define $|\begin{array}{l}{a}&{c}\\{b}&{d}\end{array}|=ad-bc$. For example, $|\begin{array}{l}{1}&{3}\\{2}&{4}\end{array}|=1\times 4-2\times 3=4-6=-2$. If $x$ and $y$ are integers, and satisfy $1 \lt |\begin{array}{l}{2}&{y}\\{x}&{3}\end{array}| \lt 3$, then the minimum value of $x+y$ is ____.

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