
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 solution to the equation $k-2x=3(k-2)$ with respect to $x$ is a non-negative number, and the system of inequalities with respect to $x$ $\left\{\begin{array}{l}{x-2(x-1)≤3}\\{\frac{2k+x}{3}≥x}\end{array}\right.$ has a solution, find the sum of the integer values of $k$ that satisfy the conditions.

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