
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 a geometric sequence $\{a_n\}$ with the first term $a_1=1$ and common ratio $q$, discuss the conditions under which the system of linear equations 
$$
\begin{cases} 
a_{1}x + a_{3}y = 3 \\ 
a_{2}x + a_{4}y = -2 
\end{cases}
$$
has no solution or infinitely many solutions based on different values of $q$.

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