
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 fractional equation about $x$ $\frac{a}{x-2}+\frac{2}{2-x}=3$ has a non-negative integer solution; and the number $a$ makes the quadratic function about $m$ $y=-m^{2}+\left(1-a\right)m+8$, when $m \lt -2$, $y$ increases as $m$ increases, then the sum of all values of the integer $a$ that satisfy the conditions is ____.

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