
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
We call a path Valid if
i. It only comprises of the following kind of steps:
A.  $(x, y) \rightarrow (x + 1, y + 1)$ B.  $(x, y) \rightarrow (x + 1, y - 1)$ ii. It never goes below the x-axis.
Let  $M(n)$   = set of all valid paths from  $(0,0) $ , to  $(2n,0)$ , where  $n$  is a natural number. 
Consider a Valid path  $T \in M(n)$ . 
Denote  $\phi(T) = \prod_{i=1}^{2n} \mu_i$ ,
where  $\mu_i$ = 
a)  $1$ , if the  $i^{th}$  step is  $(x, y) \rightarrow (x + 1, y + 1)$ b)  $y$ ,  if the  $i^{th} $  step is  $(x, y) \rightarrow (x + 1, y - 1)$  
Now Let  $f(n) =\sum _{T \in M(n)} \phi(T)$ . Evaluate the number of zeroes at the end in the decimal expansion of  $f(2021)$ 

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