
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
A *9-cube* is a nine-dimensional hypercube (and hence has  $2^9$  vertices, for example). How many five-dimensional faces does it have?

(An  $n$  dimensional hypercube is defined to have vertices at each of the points  $(a_1,a_2,\cdots ,a_n)$  with  $a_i\in \{0,1\}$  for  $1\le i\le n$ )

*Proposed by Evan Chen*

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