
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
The classrooms at MIT are each identified with a positive integer (with no leading zeroes). One day, as President Reif walks down the Infinite Corridor, he notices that a digit zero on a room sign has fallen off. Let \( N \) be the original number of the room, and let \( M \) be the room number as shown on the sign.

The smallest interval containing all possible values of \( \frac{M}{N} \) can be expressed as \( \left[\frac{a}{b}, \frac{c}{d}\right) \) where \( a, b, c, d \) are positive integers with \( \operatorname{gcd}(a, b) = \operatorname{gcd}(c, d) = 1 \). Compute \( 1000a + 100b + 10c + d \).

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