Ifelse: combine 3 categories into one

Using this data.frame

Data

df <- read.table(text = c("
ID  cat1    cat2    cat3
site1   High    High    High
site1   High    High    Medium
site1   High    High    Low
site1   High    Medium  High
site1   High    Medium  Medium
site1   High    Medium  Low
site1   High    Low High
site1   High    Low Medium
site1   High    Low Low
site1   Medium  High    High
site1   Medium  High    Medium
site1   Medium  High    Low
site1   Medium  Medium  High
site1   Medium  Medium  Medium
site1   Medium  Medium  Low
site1   Medium  Low High
site1   Medium  Low Medium
site1   Medium  Low Low
site1   Low High    High
site1   Low High    Medium
site1   Low High    Low
site1   Low Medium  High
site1   Low Medium  Medium
site1   Low Medium  Low
site1   Low Low High
site1   Low Low Medium
site1   Low Low Low

"), header =T)

I want to create a new column called "new_category" based on cat1, cat2 and cat3.

I want each row in new_category to be have the common class "or word" in cat1, cat2 and cat3. If all values are different (High, Medium and Low), new_category will take the highest class (High in this case).

For example

If cat1 = High, cat2 = High, cat3= Medium, then new_category = High

If cat1 = High, cat2 = Medium, cat3= Low, then new_category = High

If cat1 = Medium, cat2 = Medium, cat3= Low, then new_category = Medium

I can do this using ifelse. However, there are many combinations of cat1 and cat2 and cat3.

Any suggestion for a faster or easier way to do that?