
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
Find all pairs of natural numbers $(m, n)$ such that the set $X = \{1, 2, \cdots, n\}$ contains $m$ triplet subsets $A_{1}, A_{2}, \cdots, A_{m}$ satisfying:
(i) Every pair of elements (i.e., every two-element subset) of $X$ is contained in some $A_{i}$ $(1 \leqslant i \leqslant m)$;
(ii) Each pair of sets $A_{1}, A_{2}, \cdots, A_{m}$ shares exactly one common element.

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