
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
If the system of inequalities about $x$ is $\left\{{\begin{array}{l}{-2({x-2})-x＜2}\\{\frac{{k-x}}{2}≥-\frac{1}{2}+x}\end{array}}\right.$ has at most $2$ integer solutions, and the solution to the one-variable linear equation about $y$ is $3\left(y-1\right)-2\left(y-k\right)=7$, determine the sum of all integers $k$ that satisfy the conditions.

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