
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 each real number  $x,$  let \[f(x)=\sum_{n\in S_x}\frac1{2^n}\] where  $S_x$  is the set of positive integers  $n$  for which  $\lfloor nx\rfloor$  is even.

What is the largest real number  $L$  such that  $f(x)\ge L$  for all  $x\in [0,1)$ ?

(As usual,  $\lfloor z\rfloor$  denotes the greatest integer less than or equal to  $z.$ 

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