
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
What is the sum of all three-digit numbers that are divisible by both the sum and the product of their digits with the divisibility constraint based on $ab \mid 100a + 10b + c$ and $a+b+c \mid 100a + 10b + c$, where neither $a$ (hundreds place), $b$ (tens place), nor $c$ (units place) is zero?

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