
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
Let  $(a_n)_n\geq 0$  and  $a_{m+n}+a_{m-n}=\frac{1}{2}(a_{2m}+a_{2n})$  for every  $m\geq n\geq0.$  If  $a_1=1,$  then find the value of  $a_{2007}.$ 

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