
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
Given random variable $X\sim N(\mu, \sigma^2)$ with probability density function $f(x)=\frac{1}{{\sigma\sqrt{2\pi}}}{e^{-\frac{{{{(x-\mu)}^2}}}{{2{\sigma^2}}}}}$, and the maximum value at $x=\sqrt{2}$ is $\frac{1}{{2\sqrt{\pi}}}$, and given $P\left(\mu -\sigma \leq X\leq \mu +\sigma \right)=0.6827$ and $P\left(\mu -2\sigma \leq X\leq \mu +2\sigma \right)=0.9545$, determine the probability $P\left(X \gt 0\right)$.

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