
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
For any positive integer $n$, the double factorial of $n$, denoted as $n!!$, is defined as follows:  
When $n$ is an even number, $n!! = n(n-2)(n-4)…6\cdot4\cdot2$.  
When $n$ is an odd number, $n!! = n(n-2)(n-4)…5\cdot3\cdot1$.  
Determine the number of correct propositions among: $(2011!!)(2010!!) = 2011!$, $2010!! = 2\cdot1005!$, $(2010!!)(2010!!) = 2011!$, The units digit of $2011!!$ is 5.

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