
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
Say that a polynomial with real coefficients in two variable,  $ x,y,$  is *balanced* if the average value of the polynomial on each circle centered at the origin is  $ 0.$  The balanced polynomials of degree at most  $ 2009$  form a vector space  $ V$  over  $ \mathbb{R}.$  Find the dimension of  $ V.$ 

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