
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
Let  $X_1$ ,  $X_2$ , ...,  $X_{2012}$  be chosen independently and uniformly at random from the interval  $(0,1]$ . In other words, for each  $X_n$ , the probability that it is in the interval  $(a,b]$  is  $b-a$ . Compute the probability that  $\lceil\log_2 X_1\rceil+\lceil\log_4 X_2\rceil+\cdots+\lceil\log_{1024} X_{2012}\rceil$  is even. (Note: For any real number  $a$ ,  $\lceil a \rceil$  is defined as the smallest integer not less than  $a$ .)

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