# Big, Bigger, Biggest!

**Calculus**Level 5

Let \(x=10^{100}, y=10^{x}, z=10^{y}\) and let

\( a_{1}=x!, a_{2}=x^{y}, a_{3}=y^{x}, a_{4}=z^{x}, a_{5}=e^{xyz}, a_{6}=z^{\frac{1}{y}}, a_{7}= y^{\frac{z}{x}}\).

You might want to use the Stirling's approximation of \(n! \approx \sqrt{2\pi}n^{(n+\frac{1}{2})}e^{-n}\) for large \(n\).

Arrange \(a_{1}, a_{2}, a_{3}, a_{4}, a_{5}, a_{6}, a_{7}\) in increasing order and catenate the index number of your sequence as the answer.

For example, if you think \(a_{1}<a_{2}<a_{3}<a_{4}<a_{5}<a_{6}<a_{7}\), then input \(1234567\) as the answer.

###### This problem is a shameless ripoff of STEP (Sixth Term Examination Papers) 1999 Pure Mathematics- Question 1.

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