
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
Consider the following modified algorithm for binary search, which we will call  $\textit{weighted binary search}$ :

\begin{tabular}{l}
01: \textbf{FUNCTION} SEARCH( $L$ , value) 
02: $\qquad$   hi  $\leftarrow$   $\operatorname{len}(L) - 1$  
03: $\qquad$   lo  $\leftarrow$  0 
04: $\qquad$   \textbf{WHILE} hi  $\geq$  lo 
05: $\qquad\qquad$  guess  $\leftarrow$   $\lfloor w \cdot\text{lo} + (1-w) \cdot \text{hi}\rfloor$  
06: $\qquad\qquad$  mid  $\leftarrow$   $L[\text{guess}]$  
07: $\qquad\qquad$  \textbf{IF} mid  $> \text{value}$  
08:  $\qquad\qquad\qquad$  hi  $\leftarrow$   $\text{guess} - 1$  
09:  $\qquad\qquad$  \textbf{ELSE IF} mid  $< \text{value}$  
10:  $\qquad\qquad\qquad$  lo  $\leftarrow$   $\text{guess} + 1$  
11:  $\qquad\qquad$  \textbf{ELSE} 
12:  $\qquad\qquad\qquad$  \textbf{RETURN} guess 
13: $\qquad$  \textbf{RETURN} -1 (not found) 
\end{tabular}

Assume  $L$  is a list of the integers  $\{1,2,\ldots,100\}$ , in that order. Further assume that accessing the  $k$ th index of  $L$  costs  $k+1$  tokens (e.g.  $L[0]$  costs  $1$  token). Let  $S$  be the set of all  $w\in[\tfrac12,1)$  which minimize the average cost when  $\texttt{value}$  is an integer selected at random in the range  $[1,50]$ . Given that  $S=\left(x,\tfrac {74}{99}\right]$ , determine  $x$ .

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