
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
Given that $m$ and $n$ are positive integers, and $m \geq n$. A rectangular solid made up of $5mn$ unit cubes has its length, width, and height in the order of $m$, $n$, and $5$, respectively. If three faces of this rectangular solid that meet at a certain vertex are painted, and exactly half of the unit cubes have no faces painted, then the ordered pair $(m, n) = $.

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