
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  $S$  denote the set of words  $W = w_1w_2\ldots w_n$  of any length  $n\ge0$  (including the empty string  $\lambda$ ), with each letter  $w_i$  from the set  $\{x,y,z\}$ . Call two words  $U,V$  *similar* if we can insert a string  $s\in\{xyz,yzx,zxy\}$  of three consecutive letters somewhere in  $U$  (possibly at one of the ends) to obtain  $V$  or somewhere in  $V$  (again, possibly at one of the ends) to obtain  $U$ , and say a word  $W$  is *trivial* if for some nonnegative integer  $m$ , there exists a sequence  $W_0,W_1,\ldots,W_m$  such that  $W_0=\lambda$  is the empty string,  $W_m=W$ , and  $W_i,W_{i+1}$  are similar for  $i=0,1,\ldots,m-1$ . Given that for two relatively prime positive integers  $p,q$  we have
\[\frac{p}{q} = \sum_{n\ge0} f(n)\left(\frac{225}{8192}\right)^n,\]where  $f(n)$  denotes the number of trivial words in  $S$  of length  $3n$  (in particular,  $f(0)=1$ ), find  $p+q$ .

*Victor Wang*

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