
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 card trick is performed using a standard deck of cards. All face cards (jacks, queens, and kings) are counted as tens. 

1. Look at the top card of the deck (for example, a seven), place it face down on the table, and continue counting aloud in order: "Eight, nine, ten..." up to 12, placing another card from the deck on top for each count. In this example, if the bottom card is a seven, a pile of 6 cards will form on the table.

2. Look at the now top card of the remaining deck (for example, a face card, now counted as ten), place it face down on the table, and continue counting in order up to 12, placing one card on top from the deck for each count. This time, a pile of 3 cards will form on the table.

3. Repeat this process until the entire deck is exhausted. If at the end there are not enough cards to complete a stack (counting to 12), set this incomplete pile aside.

Report how many stacks you have formed and how many cards you set aside. 

I will then calculate the sum of the values of the bottom cards in all the stacks using the formula: 
\[ 13 \times (\text{number of stacks} - 4) + \text{number of set-aside cards} \]
For example, if there are 6 stacks and 5 set-aside cards, the sum will be \( 13 \times (6 - 4) + 5 = 31 \).

Why does this calculation work?

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