
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 following rules:
From any lattice point \((a,b)\), the particle 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.
How many different paths can the particle take from \((0,0)\) to \((6,6)\)?

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