Is it permissible for an Imam who doubts the number of Rak'ahs to rely on the affirmation and silence of the congregants, and in this case, should he perform the prostration of forgetfulness (sujud al-sahw) after the salaam or before it? What is the Maliki and Shafi'i madhhab regarding the position of the prostration of forgetfulness, along with the ذكر الأدلة (mention of evidence)?

There is no evidence for the Imam to base his prayer on the method mentioned by the questioner. The Sunnah is for the Imam to base it on certainty or on the preponderance of his opinion. The well-known view in the Hanbali school differentiates between the Imam and the individual praying alone. The individual praying alone bases his prayer on certainty, while the Imam bases it on the preponderance of his opinion. The prostrations of forgetfulness (sujood al-sahw) are performed after the salutation in two cases: if he gave the salutation after a deficiency of one or more rak'ahs, and if he based his prayer on the preponderance of his opinion. Otherwise, the prostrations of forgetfulness are performed before the salutation. As for the Shafi'i school, all instances of prostration are, in their view, preferably performed before the salutation.