
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 palindromic number is a positive integer that reads the same from left to right as it does from right to left. Examples include 22, 121, 3443, 94249, etc. It's clear that there are 9 two-digit palindromic numbers: 11, 22, 33, ..., 99. There are 90 three-digit palindromic numbers: 101, 111, 121, ..., 191, 202, ..., 999. Determine the following:
(1) How many four-digit palindromic numbers are there?
(2) How many palindromic numbers are there with \(2n+1\) (where \(n \in \mathbb{N}^*\)) digits?

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