
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 that the greatest integer function $[z]$ is defined, and $x$ and $y$ satisfy the simultaneous equations  
\begin{align*} y&=2[x]+3 \\ y&=3[x-2]+5, \end{align*} if $x$ is not an integer, determine the range of $x+y$.

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