Remainder of a power tower under modulo $2013$

Question

I have an expression like this:

$$\left(\large 6000^{5999^{5998^{5997^{{\ldots^{1}}}}}}\right)\bmod 2013$$

Then which method should I use to solve it? Please provide the method not the answer.

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Editor's Note: Note that this is a power tower with different values and not the same value as with general tetration. Also, don't confuse tetration with exponentiation. Both are completely different.

Answer 1

Hint: Work out the answer modulo $3$, $11$ and $61$ (the prime factors of $2013$) separately. Use Euler's theorem to climb the exponent ladder. In the end combine the answers by using the Chinese remainder theorem.