
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
An ordering of a set of \( n \) elements is a bijective map between the set and \(\{1,2, \ldots, n\}\). Call an ordering \(\rho\) of the 10 unordered pairs of distinct integers from the set \(\{1,2,3,4,5\}\) admissible if, for any \(1 \leq a<b<c \leq 5\), either \(p(\{a, b\})<p(\{a, c\})<p(\{b, c\})\) or \(p(\{b, c\})<p(\{a, c\})<p(\{a, b\})\). Find the total number of admissible orderings.

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