Analysis: Direct proof is not easy by hand, so we use the idea of specialization to examine the characteristics of the terms of the sequence.

From the given conditions, we can deduce that the first 9 terms of the sequence $\left\{a_{n}\right\}$ are $1,2,7,29,22,23,49,26,-17$. It can be observed that:
(1) The first 9 terms are not multiples of 3;
(2) The first 9 terms are not multiples of 4;
(3) Modulo 3, the first 9 terms become $1,2,1,2,1,2$, $1,2,1$, which is clearly a periodic function;
(4) Modulo 4, the first 9 terms become $1,2,3,1,2,3$, $1,2,3$, which is also a periodic function.

Conjecture: These 4 properties hold for all terms of the sequence $\left\{a_{n}\right\}$, thus generating 4 strengthened propositions. They can all be proven using mathematical induction, and the first two can also be proven using the extreme principle.

Proof (3): Clearly, $a_{1} \equiv 1(\bmod 3), \bar{a}_{2}=$ $2(\bmod 3)$.
Assume $a_{2 k-1}=1(\bmod 3), a_{2 k} \equiv 2(\bmod 3)$.
(1) If $a_{2 k-1} a_{2 k}$ is even, then $a_{2 k+1}=5 a_{2 k}-$ $3 a_{2 k-1} \equiv 5 \times 2-3 \times 1 \equiv 1(\bmod 3)$;
(2) If $a_{2 k-1} a_{2 k}$ is odd, then $a_{2 k+1}=a_{2 k}-$ $a_{2 k-1} \equiv 2-1 \equiv 1(\bmod 3)$.
Similarly, it can be proven that $a_{2 k+2} \equiv 2(\bmod 3)$.