
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 system of linear equations in two variables $x$ and $y$: $\left\{\begin{array}{c}x-y=3-a\\ x+2y=5a\end{array}\right.$ ($a$ is a real number).<br/>$(1)$ If the solutions of the system always satisfy $y=a+1$, find the value of $a$;<br/>$(2)$ It is known that the solutions of the system are also solutions of the equation $bx+3y=1$ ($b$ is a real number, $b\neq 0$ and $b\neq -6)$<br/>① Investigate the relationship between the real numbers $a$ and $b$;<br/>② If both $a$ and $b$ are integers, find the maximum and minimum values of $b$.

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