
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 non-zero vectors $\overrightarrow{a}=x(\overrightarrow{{e}_{1}}+\overrightarrow{{e}_{2}})$, $\overrightarrow{b}=\overrightarrow{{e}_{1}}+y\overrightarrow{{e}_{2}}$, where $\overrightarrow{{e}_{1}}$, $\overrightarrow{{e}_{2}}$ are a pair of non-collinear vectors. Find the values of real numbers $x$ and $y$ that make the directions of $\overrightarrow{a}$ and $\overrightarrow{b}$ opposite.

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