Each problem is new in different ways. And mechanically applying specialized tools often doesn’t take you far. So beyond specialized tools, you need meta-skills.
The top meta-skill is learning. Immersing yourself in the area you are thinking about will help you solve problems better and quicker. Learning more broadly helps as well—it enables you to connect dots arrayed in unusual patterns.
Only second to learning is writing. Writing works because it is an excellent tool for thinking. Humans have limited memories, finite processing capacity, are overconfident, and are subject to ‘passions’ of the moment that occlude thinking. Writing reduces the malefic effects of these deficiencies.
By incrementally writing things down, you no longer have to store everything in the brain. Having a written copy also means that you can repeatedly go over the contents, which makes focusing on each of the points easier. But having something written also means you can `scan’ more quickly. Writing down, thus, also allows you to mix and match and form new combinations more easily.
Just as writing overcomes some of the limitations of our memory, it also improves our computational power. Writing allows us to overcome finite processing capacity by spreading the computation over time—run Intel 8088 for a long time, and you can solve reasonably complex problems.