
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 $\overrightarrow{a}=({1,-1})$, $\overrightarrow{b}=({-2,3})$, $\overrightarrow{c}=({-6,8})$.

$(1)$ Let $\overrightarrow{c}=x\overrightarrow{a}+y\overrightarrow{b}$, find the values of $x$ and $y$.

$(2)$ When the angle between $({k\overrightarrow{a}+2\overrightarrow{b}})$ and $\overrightarrow{c}$ is acute, find the range of values for $k$.

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