
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 \( A = \overline{abcba} \) be a five-digit palindromic number, where \( a \neq 0 \). If \( 1 \leq a \leq 8 \), then the last digit of the number \( A+11 \) will be \( a+1 \), and thus the first digit of \( A+11 \) must also be \( a+1 \). This is possible only with carrying over the unit place, i.e., when \( b = c = 9 \). Then \( A+11 \) is a palindromic number of the form \( (a+1)999(a+1) \) for any \( a = 1, 2, \ldots, 8 \). The case \( a = 9 \) is not possible because \( A+11 \) ends in zero and, due to symmetry, must start with zero. However, a number cannot start with zero. 

The total number of solutions is equal to the number of possible values for \( a \), which is eight.

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