
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 in terms of $x$ and $y$: $\left\{\begin{array}{c}x-y=2a+1\\ 2x+3y=9a-8\end{array}\right.$, where $a$ is a real number.<br/>$(1)$ If $x=y$, find the value of $a$;<br/>$(2)$ If the point $\left(x,y\right)$ is in the fourth quadrant and is equidistant from the $x$-axis and $y$-axis, find the value of $a$.

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