
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
For a given value  $t$ , we consider number sequences  $a_1, a_2, a_3,...$  such that  $a_{n+1} =\frac{a_n + t}{a_n + 1}$  for all  $n \ge 1$ .

(a) Suppose that  $t = 2$ . Determine all starting values  $a_1 > 0$  such that  $\frac43 \le a_n \le \frac32$  holds for all  $n \ge 2$ .

(b) Suppose that  $t = -3$ . Investigate whether  $a_{2020} = a_1$  for all starting values  $a_1$  different from  $-1$  and  $1$ .

Present the answer in LaTex format: \boxed{Your answer}
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