
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
In a game of Chomp, two players alternately take bites from a 5-by-7 grid of [unit squares](https://artofproblemsolving.com/wiki/index.php/Unit_square). To take a bite, a player chooses one of the remaining [ squares](https://artofproblemsolving.com/wiki/index.php/Square_(geometry)), then removes ("eats") all squares in the quadrant defined by the left edge (extended upward) and the lower edge (extended rightward) of the chosen square. For example, the bite determined by the shaded square in the diagram would remove the shaded square and the four squares marked by $\times.$ (The squares with two or more dotted edges have been removed form the original board in previous moves.) 
[AIME 1992 Problem 12.png](https://artofproblemsolving.com/wiki/index.php/File:AIME_1992_Problem_12.png)
The object of the game is to make one's opponent take the last bite. The diagram shows one of the many [subsets](https://artofproblemsolving.com/wiki/index.php/Subset) of the [set](https://artofproblemsolving.com/wiki/index.php/Set) of 35 unit squares that can occur during the game of Chomp. How many different subsets are there in all? Include the full board and empty board in your count.

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