
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
Let $\overline{a b c d}$ be a four-digit number where the first two digits form the two-digit number $\overline{a b}$ and the last two digits form the two-digit number $\overline{c d}$. If it satisfies the conditions that $\overline{a b} \neq \overline{c d}$, $\overline{a b}$ divides $\overline{c d}$, and $\overline{c d}$ divides $\overline{a b c d}$, we call such a four-digit number a "lucky number."

(1) Determine the total number of "lucky numbers."

(2) Find the largest odd "lucky number."

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