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Advanced search form with multiple custom fields

I’m working in a site that allows to search for lawsuits about
medical malpractice, the client wants to have a search form almost equal to the one in this site:

http://psychcrime.org/database/

All the information is stored in custom post types with custom fields, including the lawsuit, hospital and the doctor record.

I looked around for plugins but most of the advanced plugins I found are for custom taxonomies filters and not custom fields and some have the ability to search in custom fields but with only one input in the form instead of separated inputs for each possible field to search.

Is there a plugin that can achieve this layout and behaviour?
If there is not what could be my best approach in creating this search form? I have never modified the wordpress search before.

We are using advanced custom fields plugin.