
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 "Hishgad" lottery ticket contains the numbers  $1$  to  $mn$ , arranged in some order in a table with  $n$  rows and  $m$  columns. It is known that the numbers in each row increase from left to right and the numbers in each column increase from top to bottom. An example for  $n=3$  and  $m=4$ :

[asy]
size(3cm);
Label[][] numbers = {{" $1$ ", " $2$ ", " $3$ ", " $9$ "}, {" $4$ ", " $6$ ", " $7$ ", " $10$ "}, {" $5$ ", " $8$ ", " $11$ ", " $12$ "}};
for (int i=0; i<5;++i) {
draw((i,0)--(i,3));
}
for (int i=0; i<4;++i) {
draw((0,i)--(4,i));
}
for (int i=0; i<4;++i){
for (int j=0; j<3;++j){
label(numbers[2-j][i], (i+0.5, j+0.5));
}}
[/asy]
When the ticket is bought the numbers are hidden, and one must "scratch" the ticket to reveal them. How many cells does it always suffice to reveal in order to determine the whole table with certainty?

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