
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
Seven cards numbered  $1$  through  $7$  lay stacked in a pile in ascending order from top to bottom ( $1$  on top,  $7$  on bottom). A shuffle involves picking a random card *of the six not currently on top*, and putting it on top. The relative order of all the other cards remains unchanged. Find the probability that, after  $10$  shuffles,  $6$  is higher in the pile than  $3$ .

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