
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  $f(x) = 2x^2 + x - 1, f^{0}(x) = x$ , and  $f^{n+1}(x) = f(f^{n}(x))$  for all real  $x>0$  and  $n \ge 0$  integer (that is,  $f^{n}$  is  $f$  iterated  $n$  times).

a) Find the number of distinct real roots of the equation  $f^{3}(x) = x$ b) Find, for each  $n \ge 0$  integer, the number of distinct real solutions of the equation  $f^{n}(x) = 0$ 

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