
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 female student walks from home to school every day on streets oriented in east-west and north-south directions. Her home is at $O(0,0)$, and her school is at $P(9,8)$. She always chooses the shortest path, thus she doesn't walk from east to west or north to south. 
(1) How many different possible paths are there?
(2) If her best friend lives at $A(3,2)$ and she always picks up her friend before heading to school, how many different paths are there in this case?
(3) If there is a hazardous zone at the intersection $B(6,5)$ (e.g., a gas leak) prohibiting passage, and she must pick up her friend at $A$ before going to school while avoiding $B$, how many different paths are there?
(4) If the hazardous zone is the street segment $B C(B C$ between the open line segment $C(8,5))$, how many different paths are available?

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