
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 permutations \( \left(a_{0}, a_{1}, \ldots, a_{2022}\right) \) of \( (0, 1, \ldots, 2022) \) such that:
- \( a_{2022} = 625 \),
- For each \( 0 \leq i \leq 2022, a_{i} \geq \frac{625 i}{2022} \),
- For each \( 0 \leq i \leq 2022,\left\{a_{i}, \ldots, a_{2022}\right\} \) is a set of consecutive integers (in some order).

The number of such permutations can be written as \( \frac{a!}{b!c!} \) for positive integers \( a, b, c \), where \( b > c \) and \( a \) is minimal. Compute \( 100a + 10b + c \).

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