ITMC 2018 Grade 7 National Preliminary set01 Question 16

Exponentiation $b^n$ has base $b$ and exponent $n$. For positive integer $n$, it means multiplying $n$ copies of $b$. For example, $3^2=9$, with base 3 and exponent 2. A tower is evaluated from the top: $3^{2^4}=3^{16}$ has base 3 and exponent $2^4=16$, not base $3^2$ and exponent 4. The latter requires parentheses: $(3^2)^4=3^8$.

How many integers $x$ satisfy $x^{x^x}=(x^x)^x$?