
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 the system of linear equations in two variables $x$ and $y$: $\left\{\begin{array}{l}{2x-3y=-2}\\{x-2y=k}\end{array}\right.$, where the solution satisfies $x-y \lt 0$. Find:<br/>$(1)$ the range of values for $k$;<br/>$(2)$ under the condition in $(1)$, if the solution to the inequality $\left(2k+1\right)x \lt 2k+1$ is $x \gt 1$, determine the integer values of $k$ that satisfy the condition.

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