
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 such that from any lattice point $(a,b)$, it may only move to $(a+1,b)$, $(a,b+1)$, or $(a+1,b+1)$. There are no right angle turns in the particle's path. Determine the number of different paths the particle can take from $(0,0)$ to $(6,6)$.

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