
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
The solution to the fractional equation $\frac{3x-a}{x-3}+\frac{x+1}{3-x}=1$ with respect to $x$ is a positive number, and the solution set of the inequality with respect to $y$ $\left\{\begin{array}{l}{y+9＜2(y+2)}\\{\frac{2y-a}{3}≥1}\end{array}\right.$ is $y \gt 5$. The sum of all integers $a$ that satisfy the conditions is ____.

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