
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
Let  $P(x)=x+1$  and  $Q(x)=x^2+1.$  Consider all sequences  $\langle(x_k,y_k)\rangle_{k\in\mathbb{N}}$  such that  $(x_1,y_1)=(1,3)$  and  $(x_{k+1},y_{k+1})$  is either  $(P(x_k), Q(y_k))$  or  $(Q(x_k),P(y_k))$  for each  $k.  $  We say that a positive integer  $n$  is nice if  $x_n=y_n$  holds in at least one of these sequences. Find all nice numbers.

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