
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  $ P$  be the product of all non-zero digits of the positive integer  $ n$ . For example,  $ P(4) \equal{} 4$ ,  $ P(50) \equal{} 5$ ,  $ P(123) \equal{} 6$ ,  $ P(2009) \equal{} 18$ . 
Find the value of the sum: P(1) + P(2) + ... + P(2008) + P(2009).

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