
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
Given vectors  $v_1, \dots, v_n$  and the string  $v_1v_2 \dots v_n$ ,
we consider valid expressions formed by inserting  $n-1$  sets of balanced parentheses and  $n-1$  binary products,
such that every product is surrounded by a parentheses and is one of the following forms:

1. A "normal product''  $ab$ , which takes a pair of scalars and returns a scalar, or takes a scalar and vector (in any order) and returns a vector. 

2. A "dot product''  $a \cdot b$ , which takes in two vectors and returns a scalar. 

3. A "cross product''  $a \times b$ , which takes in two vectors and returns a vector. 

An example of a *valid* expression when  $n=5$  is  $(((v_1 \cdot v_2)v_3) \cdot (v_4 \times v_5))$ , whose final output is a scalar. An example of an *invalid* expression is  $(((v_1 \times (v_2 \times v_3)) \times (v_4 \cdot v_5))$ ; even though every product is surrounded by parentheses, in the last step one tries to take the cross product of a vector and a scalar. 

Denote by  $T_n$  the number of valid expressions (with  $T_1 = 1$ ), and let  $R_n$ denote the remainder when  $T_n$  is divided by  $4$ .
Compute  $R_1 + R_2 + R_3 + \ldots + R_{1,000,000}$ .

*Proposed by Ashwin Sah*

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