
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 particle moves in the Cartesian plane according to the updated rules:
From any lattice point $(x,y),$ the particle may move to $(x+1,y), (x,y+1),$ or $(x-1,y).$
The particle's path can now include right-angle turns.
How many different paths can the particle take from $(0,0)$ to $(3,3)$?

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