
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 three-digit number has a tens digit equal to the average of the hundreds digit and the units digit, we call this three-digit number a "diligent number." For example: $630$, $123$. The largest "diligent number" is ______. If the three-digit number $\overline{abc}$ is a "diligent number," and the sum of the digits is greater than $7$ and less than $10$, and the hundreds digit $a$ satisfies the condition that the one-variable quadratic equation $\left(a-5\right)x^{2}+2ax+a-8=0$ has real roots, then the sum of all "diligent numbers" that meet the conditions is ______.

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