
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 from $(0,0)$ to $(6,6)$ following these rules:
1. Allowed moves from any lattice point $(a,b)$ are to $(a+1,b), (a,b+1),$ or $(a+1,b+1)$.
2. The particle’s path cannot include right angle turns.
3. No diagonal move can be immediately followed by a move in the vertical or horizontal directions that form a right angle with the previous move.
How many distinct paths can the particle take?

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