
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 \([x]\) denote the largest integer not exceeding \(x\). Find the value of \(\left[\log_2 1\right] + \left[\log_2 2\right] + \left[\log_2 3\right] + \cdots + \left[\log_2 2009\right]\).

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