
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
We have some cards that have the same look, but at the back of some of them is written  $0$  and for the others  $1$ .(We can't see the back of a card so we can't know what's the number on it's back). we have a machine. we give it two cards and it gives us the product of the numbers on the back of the cards. if we have  $m$  cards with  $0$  on their back and  $n$  cards with  $1$  on their back, at least how many times we must use the machine to be sure that we get the number  $1$ ? (15 points)

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