
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
In the diagram below, how many distinct paths are there from January 1 to December 31, moving from one adjacent dot to the next either to the right, down, or diagonally down to the right?

[asy]
size(340);
int i, j;
for(i = 0; i<10; i = i+1) {
for(j = 0; j<5; j = j+1) {
if(10*j + i == 11 || 10*j + i == 12 || 10*j + i == 14 || 10*j + i == 15 || 10*j + i == 18 || 10*j + i == 32 || 10*j + i == 35 || 10*j + i == 38 ) { }
else{ label(" $*$ ", (i,j));}
}}
label(" $\leftarrow$ "+"Dec. 31", (10.3,0));
label("Jan. 1"+" $\rightarrow$ ", (-1.3,4));[/asy]

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