
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

For letters \( a \sim z \), encode them as $(\mathbf{a}=1, \mathbf{b}=2, \ldots, \mathbf{z}=26)$. Each English word (assuming all letters in the word are lowercase) can be assigned a product \(\mathbf{p}\) calculated from these letter codes. For example, for the word "good," its corresponding \(\mathbf{p}\) value is $7 \times 15 \times 15 \times 4 = 6300$ (because \(g = 7, o = 15, d = 4\)). If a composite number cannot be expressed as the \(\mathbf{p}\) value of any word (regardless of whether the word has meaning), such a composite number is called a "middle ring number". What is the smallest three-digit "middle ring number"?

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