
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  $K$ , define a sequence,  $\{a_n\}_n$ , as following  $a_1=K$ , \[ a_{n+1} = \{ \begin{array} {cc} a_n-1 , & \mbox{ if } a_n \mbox{ is even}  \frac{a_n-1}2 , & \mbox{ if } a_n \mbox{ is odd} \end{array},  \] for all  $n\geq 1$ . 

Find the smallest value of  $K$ , which makes  $a_{2005}$  the first term equal to 0.

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