
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 $(a+1,b+1).$
The particle starts at $(0,0)$ and finishes at $(6,6)$. Calculate the total number of distinct paths the particle can take, assuming each step is toward one of these allowed points.

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