
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
In a topological space, a function $f(x)$ whose graph satisfies certain conditions is said to be a "fixed-point" function if there exists a point $x_{0}$ such that $f(x_{0})=x_{0}$, where $x_{0}$ is called the fixed point of the function. By analogy, a fixed point $x_{0}$ satisfying ${f'}(x_{0})=x_{0}$ is called a double fixed point of $f(x)$. Among the following functions, ① $f(x)=x^{3}-x\sin x$; ② $f(x)={e^x}-\frac{1}{x}$; ③ $f(x)=\frac{{{e^x}+{e^{-x}}}}{2}-1$, the functions that have double fixed points are ______. (Fill in the blank with the code of the functions you think are correct)

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