
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  $(F_n)$  be the sequence defined recursively by  $F_1=F_2=1$  and  $F_{n+1}=F_n+F_{n-1}$  for  $n\geq 2$ . Find all pairs of positive integers  $(x,y)$  such that  $$ 5F_x-3F_y=1. $$  

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