
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
Expand the trinomial $(x^2+x+1)^n$. For $n=1, 2, 3, \ldots$, the expansion yields the generalized Pascal's triangle as shown below: 
$$(x^2+x+1)^0 = 1$$
$$(x^2+x+1)^1 = x^2 + x + 1$$
$$(x^2+x+1)^2 = x^4 + 2x^3 + 3x^2 + 2x + 1$$
$$(x^2+x+1)^3 = x^6 + 3x^5 + 6x^4 + 7x^3 + 6x^2 + 3x + 1$$
$$(x^2+x+1)^4 = x^8 + 4x^7 + 10x^6 + 16x^5 + 19x^4 + 16x^3 + 10x^2 + 4x + 1$$
...
Observing the relationships between the coefficients of the polynomials, we can construct the generalized Pascal's triangle as shown. The construction rule is: the first row is 1, and each number in the following rows is the sum of the three numbers above it (if any of these three numbers do not exist, count them as 0), and the $k^{th}$ row consists of $2k+1$ numbers. If in the expansion of $(1+ax)(x^2+x+1)^5$, the coefficient of the $x^8$ term is 75, then the real number $a$ is __________.

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