
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
Given a string of at least one character in which each character is either A or B, Kathryn is allowed to make these moves:


    
- she can choose an appearance of A, erase it, and replace it with BB, or
- she can choose an appearance of B, erase it, and replace it with AA.


Kathryn starts with the string A. Let  $a_n$  be the number of strings of length  $n$  that Kathryn can reach using a sequence of zero or more moves. (For example,  $a_1=1$ , as the only string of length 1 that Kathryn can reach is A.) Then  $\sum_{n=1}^{\infty} \frac{a_n}{5^n} = \frac{m}{n}$ , where  $m$  and  $n$  are positive integers with  $\gcd(m,n)=1$ . Compute  $100m+n$ .

*Proposed by Luke Robitaille*

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