
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
Dominic chooses a two-digit positive integer $M$ and writes both its base-4 and base-7 representations on a whiteboard. He then, treating these numbers as if they were in base-10, adds them to get an integer $P$. Determine for how many values of $M$ the two rightmost digits of $P$, in order, are the same as those of $3M$.
A) 10
B) 15
C) 20
D) 25
E) 30

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