
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  $f(x) = x-\tfrac1{x}$ , and define  $f^1(x) = f(x)$  and  $f^n(x) = f(f^{n-1}(x))$  for  $n\ge2$ . For each  $n$ , there is a minimal degree  $d_n$  such that there exist polynomials  $p$  and  $q$  with  $f^n(x) = \tfrac{p(x)}{q(x)}$  and the degree of  $q$  is equal to  $d_n$ . Find  $d_n$ .

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