
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

A number is said to be TOP if it has 5 digits and when the product of the 1st and 5th digits is equal to the sum of the 2nd, 3rd, and 4th digits. For example, 12,338 is TOP because it has 5 digits and $1 \cdot 8 = 2 + 3 + 3$.

a) What is the value of $a$ such that $23,4a8$ is TOP?

b) How many TOP numbers end with 2 and start with 1?

c) How many TOP numbers start with 9?

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