
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
Given an integer  $ m$ , define the sequence  $ \left\{a_{n}\right\}$  as follows:
\[ a_{1}\equal{}\frac{m}{2},\ a_{n\plus{}1}\equal{}a_{n}\left\lceil a_{n}\right\rceil,\textnormal{ if }n\geq 1\]
Find all values of  $ m$  for which  $ a_{2007}$  is the first integer appearing in the sequence.

Note: For a real number  $ x$ ,  $ \left\lceil x\right\rceil$  is defined as the smallest integer greater or equal to  $ x$ . For example,  $ \left\lceil\pi\right\rceil\equal{}4$ ,  $ \left\lceil 2007\right\rceil\equal{}2007$ .

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