
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
For problem 11 , i couldn’t find the correct translation , so i just posted the hungarian version . If anyone could translate it ,i would be very thankful .
[tip=see hungarian]Az  $X$   ́es $ Y$  valo ́s  ́ert ́eku ̋ v ́eletlen v ́altoz ́ok maxim ́alkorrel ́acio ́ja az  $f(X)$   ́es  $g(Y )$  v ́altoz ́ok korrela ́cio ́j ́anak szupr ́emuma az olyan  $f$   ́es  $g$  Borel m ́erheto ̋,  $\mathbb{R} \to \mathbb{R}$  fu ̈ggv ́enyeken, amelyekre  $f(X)$   ́es  $g(Y)$  v ́eges sz ́ora ́su ́. Legyen U a  $[0,2\pi]$  interval- lumon egyenletes eloszl ́asu ́ val ́osz ́ınu ̋s ́egi v ́altozo ́, valamint n  ́es m pozit ́ıv eg ́eszek. Sz ́am ́ıtsuk ki  $\sin(nU)$   ́es  $\sin(mU)$  maxim ́alkorrela ́ci ́oja ́t. [/tip]
Edit:
<details><summary>Translation thanks to @tintarn</summary>The maximal correlation of two random variables  $X$  and  $Y$  is defined to be the supremum of the correlations of  $f(X)$  and  $g(Y)$  where  $f,g:\mathbb{R} \to \mathbb{R}$  are measurable functions such that  $f(X)$  and  $g(Y)$  is (almost surely?) finite.
Let  $U$  be the uniformly distributed random variable on  $[0,2\pi]$  and let  $m,n$  be positive integers. Compute the maximal correlation of  $\sin(nU)$  and  $\sin(mU)$ .
(Remark: It seems that to make sense we should require that  $E[f(X)]$  and  $E[g(Y)]$  as well as  $E[f(X)^2]$  and  $E[g(Y)^2]$  are finite.
In fact, we may then w.l.o.g. assume that  $E[f(X)]=E[g(Y)]=0$  and  $E[f(Y)^2]=E[g(Y)^2]=1$ .)</details>

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