
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 $y$: $\left\{\begin{array}{l}3x+y=k+1\\x+3y=3\end{array}\right.$

(1) If the solutions $x$ and $y$ of the system are opposite in sign, find the value of $k$.

(2) If the solutions $x$ and $y$ of the system satisfy $\left\{\begin{array}{l}x+y \lt 3\\x-y \gt 1\end{array}\right.$, find the range of values for $k$.

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