
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
A finite sequence of numbers  $(a_1,\cdots,a_n)$  is said to be alternating if   $$ a_1>a_2~,~a_2<a_3~,~a_3>a_4~,~a_4<a_5~,~\cdots $$      $$ \text{or ~}~~a_1<a_2~,~a_2>a_3~,~a_3<a_4~,~a_4>a_5~,~\cdots $$   How many alternating sequences of length  $5$  , with distinct numbers  $a_1,\cdots,a_5$  can be formed such that  $a_i\in\{1,2,\cdots,20\}$  for  $i=1,\cdots,5$  ?

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