
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
For a positive integer  $K$ , define a sequence,  $\{a_n\}$ , as following:  $a_1 = K$  and $a_{n+1} =a_n -1$  if  $a_n$  is even $a_{n+1} =\frac{a_n - 1}{2}$  if  $a_n$  is odd ,  for all  $n \ge 1$ . 
Find the smallest value of  $K$ , which makes  $a_{2005}$  the first term equal to  $0$ .





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