
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
We know the following about the functions \( g_{k}(x) \):

$$
g_{1}(x)=\frac{x^{2}}{2-x}
$$

and

$$
g_{k+1}(x)=x \frac{g_{k}(x)}{2-g_{k}(x)} \quad(k=1,2, \ldots)
$$

Determine the elements of the sequence \( g_{k}^{\prime}(1) \).

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