
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
A sequence of distinct circles  $\omega_1, \omega_2, \cdots$  is inscribed in the parabola  $y=x^2$  so that  $\omega_n$  and  $\omega_{n+1}$  are tangent for all  $n$ .  If  $\omega_1$  has diameter  $1$  and touches the parabola at  $(0,0)$ , find the diameter of  $\omega_{1998}$ .

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