
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
Given the set \( S_n = \{a_1, a_2, a_3, \ldots, a_n\} \) and \( S_m = \{0, 1, 2, 3, \ldots, m-1\} \), let \( P(S_n) \) denote the set of all subsets (including the empty set \( \emptyset \)) of \( S_n \). A mapping \( f: P(S_n) \longrightarrow S_m \) satisfies the following property: For any two elements \( X_1 \) and \( X_2 \) in \( P(S_n) \), we have
\[ 
f(X_1 \cup X_2) + f(X_1 \cap X_2) = f(X_1) + f(X_2).
\]

Find: (1) The number of such mappings \( f \) when \( f(\varnothing) = 0 \).
(2) The number of such mappings \( f \) when \( f(\emptyset) = 1 \).

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