
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 call a sequence of length  $n$  of zeros and ones a "string of length  $n$ " and the elements of the same sequence "bits". Let  $m,n$  be two positive integers so that  $m<2^n$ . Arik holds  $m$  strings of length  $n$ . Giora wants to find a new string of length  $n$  different from all those Arik holds. For this Giora may ask Arik questions of the form:

"What is the value of bit number  $i$  in string number  $j$ ?"

where  $1\leq i\leq n$  and  $1\leq j\leq m$ .
What is the smallest number of questions needed for Giora to complete his task when:**a)**  $m=n$ ?**b)**  $m=n+1$ ?

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