
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 Bernardo chooses a three-digit positive integer $N$ and writes both its base-5 and base-6 representations on a blackboard, treating the two numbers as base-10 integers, find the number of choices of $N$ for which the two rightmost digits of the sum $S$ are the same as those of $2N$.

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