
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
Consider seven-digit natural numbers in which each of the digits \(1, 2, 3, 4, 5, 6, 7\) appears exactly once.

(a) How many of these numbers have the digits from the first to the sixth in ascending order and the digits from the sixth to the seventh in descending order?

(b) How many of these numbers have the digits from the first to the fifth in ascending order and the digits from the fifth to the seventh in descending order?

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