
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
Hey, 

This problem is from the VTRMC 2006. 

3. Recall that the Fibonacci numbers  $ F(n)$  are defined by  $ F(0) \equal{} 0$ ,  $ F(1) \equal{} 1$  and  $ F(n) \equal{} F(n \minus{} 1) \plus{} F(n \minus{} 2)$  for  $ n \geq 2$ . Determine the last digit of  $ F(2006)$  (e.g. the last digit of 2006 is 6).

As, I and a friend were working on this we noticed an interesting relationship when writing the Fibonacci numbers in "mod" notation. 

Consider the following,

01 = 1 mod 10
01 = 1 mod 10 
02 = 2 mod 10 
03 = 3 mod 10 
05 = 5 mod 10 
08 = 6 mod 10 
13 = 3 mod 10 
21 = 1 mod 10 
34 = 4 mod 10 
55 = 5 mod 10 
89 = 9 mod 10

Now, consider that between the first appearance and second apperance of  $ 5 mod 10$ , there is a difference of five terms. Following from this we see that the third appearance of  $ 5 mod 10$  occurs at a difference 10 terms from the second appearance. Following this pattern we can create the following relationships.

 $ F(55) \equal{} F(05) \plus{} 5({2}^{2})$ 

This is pretty much as far as we got, any ideas?

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