
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  $n$  be a positive integer.  $f(n)$  denotes the number of  $n$ -digit numbers  $\overline{a_1a_2\cdots a_n}$ (wave numbers) satisfying the following conditions :
(i) for each  $a_i \in\{1,2,3,4\}$ ,  $a_i \not= a_{i+1}$ ,  $i=1,2,\cdots$ ;
(ii) for  $n\ge 3$ ,  $(a_i-a_{i+1})(a_{i+1}-a_{i+2})$  is negative,  $i=1,2,\cdots$ .
(1) Find the value of  $f(10)$ ;
(2) Determine the remainder of  $f(2008)$  upon division by  $13$ .

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