
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 starting from the origin $(0,0)$ and can move to any of the adjacent lattice points: $(a+1,b), (a,b+1),$ or $(a+1,b+1)$. 
However, from any point $(a,b)$, after choosing a diagonal move $(a+1, b+1)$, the particle cannot immediately make a right or upward move, i.e., no 'backtracking' on the current movement axis.
How many different paths can the particle take from $(0,0)$ to $(4,4)$?

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