
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  $F_0 = 0, F_{1} = 1$ , and for  $n \ge 1, F_{n+1} = F_n + F_{n-1}$ . Define  $a_n = \left(\frac{1 + \sqrt{5}}{2}\right)^n \cdot F_n$  . Then there are rational numbers  $A$  and  $B$  such that  $\frac{a_{30} + a_{29}}{a_{26} + a_{25}} = A + B \sqrt{5}$ . Find  $A + B$ .

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