
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 equations $\left\{\begin{array}{l}{x-y=1+3a}\\{x+y=-7-a}\end{array}\right.$ where $x$ is a non-positive number and $y$ is negative.<br/>$(1)$ Find the range of values for $a$;<br/>$(2)$ Within the range of values for $a$, for which integer value of $a$ is the solution set of the inequality $2ax+x \gt 2a+1$ $x \lt 1$.

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