
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 finite sequence of non-negative integers \( a_{1}, a_{2}, \cdots, a_{n} \) \((a_{1}=0)\) satisfies the following conditions:
(a) When \( k \geq 2 \), \( a_{k} \leq u_{k} + 1 \). Here, \( u_{2} = 0 \); for \( k \geq 3 \), \( u_{k} \) is the number of elements in the set \(\{ i \mid a_{i} < a_{i+1}, 1 \leq i \leq k-2 \} \).
(b) There do not exist indices \( i, j, k \) \((1 \leq i < j < k \leq n)\) such that \( a_{j} < a_{i} < a_{k} \).

Find the number of such sequences.

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