
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
For any finite sequence of positive integers \(\pi\), let \(S(\pi)\) be the number of strictly increasing subsequences in \(\pi\) with length 2 or more. For example, in the sequence \(\pi=\{3,1,2,4\}\), there are five increasing subsequences: \(\{3,4\}, \{1,2\}, \{1,4\}, \{2,4\}\), and \(\{1,2,4\}\), so \(S(\pi)=5\). In an eight-player game of Fish, Joy is dealt six cards of distinct values, which she puts in a random order \(\pi\) from left to right in her hand. Determine
$$
\sum_{\pi} S(\pi)
$$
where the sum is taken over all possible orders \(\pi\) of the card values.

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