
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 have a finite set \( H \) and a few of its proper subsets have been selected. It is known that

a) any two different elements of the set are in exactly one selected subset;

b) for any selected subset \( R \) and an element \( p \) not in \( R \), there are exactly \( m \) selected subsets that contain \( p \) and intersect \( R \), and exactly \( n \) selected subsets that contain \( p \) and do not intersect \( R \).

Express the number of elements in \( H \) and the number of selected subsets in terms of \( m \) and \( n \).

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