
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
Treasure was buried in a single cell of an  $M\times N$  ( $2\le M$ ,  $N$ ) grid. Detectors were brought to find the cell with the treasure. For each detector, you can set it up to scan a specific subgrid  $[a,b]\times[c,d]$  with  $1\le a\le b\le M$  and  $1\le c\le d\le N$ . Running the detector will tell you whether the treasure is in the region or not, though it cannot say where in the region the treasure was detected. You plan on setting up  $Q$  detectors, which may only be run simultaneously after all  $Q$  detectors are ready.

In terms of  $M$  and  $N$ , what is the minimum  $Q$  required to gaurantee to determine the location of the treasure?

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