
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
If the integer $a$ makes the system of inequalities about $x$ $\left\{\begin{array}{l}a-(8x+13)<0\\ x+1\leq\frac{{x+4}}{2}\end{array}\right.$ have exactly four integer solutions, and makes the fractional equation about $y$ $\frac{y}{{y-2}}+1=\frac{{ay-2}}{{2-y}}$ have an integer solution, then the sum of all integers $a$ that satisfy the conditions is ______.

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