
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
Suppose  $X$  is a random variable that takes on only nonnegative integer values, with  $E[X]=1,$   $E[X^2]=2,$  and  $E[X^3]=5.$  (Here  $E[Y]$  denotes the expectation of the random variable  $Y.$ ) Determine the smallest possible value of the probability of the event  $X=0.$ 

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