
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
Let  $E = \{1, 2, \dots , 16\}$  and let  $M$  be the collection of all  $4 \times 4$  matrices whose entries are distinct members of  $E$ . If a matrix  $A = (a_{ij} )_{4\times4}$  is chosen randomly from  $M$ , compute the probability  $p(k)$  of  $\max_i \min_j a_{ij} = k$  for  $k \in E$ . Furthermore, determine  $l \in E$  such that  $p(l) = \max \{p(k) | k \in E \}.$ 

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