
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
In the universe of Pi Zone, points are labeled with  $2 \times 2$  arrays of positive reals. One can teleport from point  $M$  to point  $M'$  if  $M$  can be obtained from  $M'$  by multiplying either a row or column by some positive real. For example, one can teleport from  $\left( \begin{array}{cc} 1 & 2  3 & 4 \end{array} \right)$  to  $\left( \begin{array}{cc} 1 & 20  3 & 40 \end{array} \right)$  and then to  $\left( \begin{array}{cc} 1 & 20  6 & 80 \end{array} \right)$ .

A *tourist attraction* is a point where each of the entries of the associated array is either  $1$ ,  $2$ ,  $4$ ,  $8$  or  $16$ .  A company wishes to build a hotel on each of several points so that at least one hotel is accessible from every tourist attraction by teleporting, possibly multiple times. What is the minimum number of hotels necessary?

*Proposed by Michael Kural*

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