
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 \( A = \{0, 1, 2, 3, 4, 5, 6, 7\} \), how many mappings \( f \) from \( A \) to \( A \) satisfy the following conditions?
1. For all \( i, j \in A \) with \( i \neq j \), \( f(i) \neq f(j) \).
2. For all \( i, j \in A \) with \( i + j = 7 \), \( f(i) + f(j) = 7 \).

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