
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
Denote  $g(k)$  as the greatest odd divisor of  $k$ . Put  $f(k) = \dfrac{k}{2} + \dfrac{k}{g(k)}$  for  $k$  even, and  $2^{(k+1)/2}$  for  $k$  odd. Define the sequence  $x_1, x_2, x_3, ...$  by  $x_1 = 1$ ,  $x_{n+1} = f(x_n)$ . Find  $n$  such that  $x_n = 800$ .

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