
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
Maria is hopping up a flight of stairs with 100 steps. At every hop, she advances some integer number of steps. Each hop she makes has fewer steps. However, the positive difference between the length of consecutive hops decreases. Let \( P \) be the number of distinct ways she can hop up the stairs. Find lower and upper bounds \( L \) and \( U \) for \( P \). If \( 0 < L \leq P \leq U \), your score will be \( \left\lfloor\frac{23}{\sqrt{U / L}}\right\rfloor \). Otherwise, your score will be 0.

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