
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
If a piece of information has $n$ ($n>1$, $n\in \mathbb{N}$) possible scenarios (each scenario is mutually exclusive), and the probabilities of these scenarios occurring are $p_1$, $p_2$, …, $p_n$ respectively, then $H=f(p_1)+f(p_2)+…+f(p_n)$ (where $f(x)=-x\log_a x$, $x\in (0,1)$) is called the information entropy of the piece of information. It is known that $f\left( \frac {1}{2}\right)= \frac {1}{2}$.  
(1) If there are 32 students in a class, and one student is selected through a random drawing to participate in an activity, try to find the magnitude of the information entropy of "who is selected";  
(2) In a competition, there are $n$ contestants (denoted as $A_1$, $A_2$, …, $A_n$) participating. If when $k=1, 2, …, n-1$, the probability of contestant $A_k$ winning the championship is $2^{-k}$, find the expression of the information entropy $H$ in terms of $n$ for "who wins the championship".

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