
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 a number $a$ makes the fractional equation about $x$ $\frac{1}{x-3}+\frac{x+a}{3-x}=1$ have a non-negative integer solution, and makes the quadratic function about $x$ $y=4x^{2}+\left(a+6\right)x-a-3$ have its axis of symmetry to the left of the $y$-axis, calculate the sum of all integers $a$ that meet the conditions.

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