Let $a$ be a positive integer.

Does there exist a positive integer $m$ other than 1 such that: $$a+m \ |\ a^m+1 \ $$ If not, what are the conditions of $a$ for $m$ to exist?

If there exist such number $m$, are there infinitely many integers $m$ that satisfy $$a+m \ |\ a^m+1 \ $$ If not, what are the conditions of $a$ so that there are infinitely many integers $m$ that satisfy the question?

This problem I think might be similar to: Given $a$, does there exist an integer $m>1$ such that: $$m \ |\ a^m+1 \ $$ I have tried to use the Lifting the Exponent Lemma and I found that if $m|a+1$, $m$ is odd and $a+1$ is not a power of $2$, then $m|a^m+1$. But for the question $a+m \ |\ a^m+1 \ $, I cannot make any progress, especially the 2nd question. Are there any ways to solve the questions ?

(Sorry, English is my second language)