
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
Concentric circles  $\Omega_1$  and  $\Omega_2$  with radii  $1$  and  $100$ , respectively, are drawn with center  $O$ . Points  $A$  and  $B$  are chosen independently at random on the circumferences of  $\Omega_1$  and  $\Omega_2$ , respectively. Denote by  $\ell$  the tangent line to  $\Omega_1$  passing through  $A$ , and denote by  $P$  the reflection of  $B$  across  $\ell$ . Compute the expected value of  $OP^2$ .

*Proposed by Lewis Chen*

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