
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
Note that the coefficients of the equation 3 and 453 are divisible by 3, so the term $5y^{2}$ is also divisible by 3, then $y=3t$, $y^{2}=9t^{2}$, where $t \in Z$. The equation becomes: 

\[ 3x^{2} + 5 \cdot 9t^{2} = 453 \] 
\[ x^{2} + 5 \cdot 3t^{2} = 151 \] 
\[ x^{2} = 151 - 15t^{2} \]

where \( x \in Z \). Let's substitute possible values of \( t \geq 0 \).

| Values of \( t \) | \( 15t^{2} \) | \( x^{2} = 151 - 15t^{2} \) | Integer value of \( x \) |
| :--- | :--- | :--- | :--- |
| \( t=0 \) | 0 | \( x^{2} = 151 \) | No |
| \( t=1 \) | 15 | \( x^{2} = 151 - 15, x^{2} = 136 \) | No |
| \( t=2 \) | 60 | \( x^{2} = 151 - 60, x^{2} = 91 \) | No |
| \( t=3 \) | 135 | \( x^{2} = 151 - 135, x^{2} = 16 \) | \( x = \pm 4 \) |
| \( t=4 \) | 240 | \( x^{2} = 151 - 240 < 0 \) | No |

Therefore, \( x= \pm 4, t= \pm 3, y= \pm 9 \).

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