
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 number, when multiplied by $18, 27, 36, 45, 54, 63, 72, 81,$ or $99$, gives a product whose first and last digits match the corresponding digits of the multiplicand, and when multiplied by $90$, gives a product whose last two digits match those of the multiplicand?

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