
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 box contains 10 red cards and 10 blue cards, each card being numbered with $1, 3, 3^{2}, \cdots, 3^{9}$ once. The total sum of the numbers on both colored cards is denoted as $S$. For a given positive integer $n$, if it is possible to select a number of cards from the box such that the sum of their numbers is exactly $n$, it is called a "$n$ selection scheme." The number of different $n$ schemes is denoted as $f(n)$. Find the value of $P=$ $\sum_{n=1}^{1000} f(n)$.

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