
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
Say that a positive integer is MPR (Math Prize Resolvable) if it can be represented as the sum of a 4-digit number MATH and a 5-digit number PRIZE.  (Different letters correspond to different digits.  The leading digits M and P can't be zero.)  Say that a positive integer is MPRUUD (Math Prize Resolvable with Unique Units Digits) if it is MPR and the set of units digits  $\{ \mathrm{H}, \mathrm{E} \}$  in the definition of MPR can be uniquely identified.  Find the smallest positive integer that is MPR but not MPRUUD.


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