
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
A certain school conducted a survey of 500 students to study the relationship between students' gender and their attitudes towards a certain activity (support and non-support). The school used a $2\times 2$ contingency table for independence testing.

|         | Male | Female |
|---------|------|--------|
| Non-support | $p$ | $40$ |
| Support | $160$ | $270$ |

After calculation, it was found that $\left(a+b\right)\left(c+d\right)\left(a+c\right)\left(b+d\right)=1.77\times 10^{9}$. The formula for $\chi ^{2}$ is given by $\chi ^{2}=\frac{n(ad-bc)^{2}}{(a+b)(c+d)(a+c)(b+d)}$.

The critical values for the chi-squared distribution are as follows:

| $P(\chi ^{2}\geqslant k)$ | $0.050$ | $0.010$ | $0.001$ |
|---------------------------|---------|---------|---------|
| $k$                       | $3.841$ | $6.635$ | $10.828$ |

$(1)$ Find the value of $p$ and calculate the numerical value of $\chi ^{2}$ (rounded to two decimal places).<br/>$(2)$ Based on the results from $(1)$, provide a correct statistical conclusion.

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