
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
Suppose that a sequence  $a_0, a_1, \ldots$  of real numbers is defined by  $a_0=1$  and \[a_n=\begin{cases}a_{n-1}a_0+a_{n-3}a_2+\cdots+a_0a_{n-1} & \text{if }n\text{ odd}a_{n-1}a_1+a_{n-3}a_3+\cdots+a_1a_{n-1} & \text{if }n\text{ even}\end{cases}\] for  $n\geq1$ . There is a positive real number  $r$  such that \[a_0+a_1r+a_2r^2+a_3r^3+\cdots=\frac{5}{4}.\] If  $r$  can be written in the form  $\frac{a\sqrt{b}-c}{d}$  for positive integers  $a,b,c,d$  such that  $b$  is not divisible by the square of any prime and  $\gcd (a,c,d)=1,$  then compute  $a+b+c+d$ .

*Proposed by Tristan Shin*

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