
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 person moves in the  $x-y$  plane moving along points with integer co-ordinates  $x$  and  $y$  only. When she is at a point  $(x,y)$ , she takes a step based on the following rules:
(a) if  $x+y$  is even she moves to either  $(x+1,y)$  or  $(x+1,y+1)$ ;
(b) if  $x+y$  is odd she moves to either  $(x,y+1)$  or  $(x+1,y+1)$ .
How many distinct paths can she take to go from  $(0,0)$  to  $(8,8)$  given that she took exactly three steps to the right  $((x,y)$  to  $(x+1,y))$ ?

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