
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
Let  $a_1, a_2, \ldots$  be a sequence of real numbers such that  $a_1=4$  and  $a_2=7$  such that for all integers  $n$ ,  $\frac{1}{a_{2n-1}}, \frac{1}{a_{2n}}, \frac{1}{a_{2n+1}}$  forms an arithmetic progression, and  $a_{2n}, a_{2n+1}, a_{2n+2}$  forms an arithmetic progression. Find, with proof, the prime factorization of  $a_{2019}$ . 

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