
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  $a$ ,  $b$ ,  $c$ ,  $d$ ,  $e$ ,  $f$  be integers selected from the set  $\{1,2,\dots,100\}$ , uniformly and at random with replacement. Set \[ M = a + 2b + 4c + 8d + 16e + 32f. \] What is the expected value of the remainder when  $M$  is divided by  $64$ ?

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